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There's a lot of headcanon on the Durarara!! alternates, especially the Shizaya ones, floating around so I've decided to make an introductory post about where the alternates originate from and the variations in alternate personalities. Keep in mind that this post is mostly about Shizuo and Izaya alternates!
ALT VS. OUTFIT
The alternates have much dispute simply because many fans think that they are the original character (Shizuo, Izaya, etc.) simply in a different outfit rather than a completely new character. While I'm not going to dispute this, the rest of this post assumes that the new outfit denotes a new personality. There is canon basis for character designs of the alternates and the source from which they appear is the basis for the fandom speculated personalities. Fanmade alternatives which have not originated from canon material (Psychedelic-420 -- Shizuo in green, Virus-138 -- Izaya in red, Psychedelic Blue -- Psyche in blue, Anti Tsugaru -- Tsugaru in red, etc.) will not be discussed here because they have no canon material basis.
ORIGINS
Official Name: Linda Linda
Fanon Name: Linda
Source: Masaomi's character song, bundled with the first DVD
Release Date: 24 February 2010
Release Position: 1

Official Name: Hachimenroppi (Range and Versatility)
Fanon Name: Roppi
Source: First Durarara!! drama CD cover, bundled with the second DVD
Release Date: 24 March 2010
Release Position: 2
Official Name: Tsugaru Kaikyou Fuyugeshiki (Tsugaru Strait - Winter Scene)
Fanon Name: Tsugaru
Source: Shizuo's character song , bundled with the third DVD
Release Date: 21 April 2010
Release Position: 3
Official Name: Psychedelic Dreams, vol. 01
Fanon Name: Psyche
Source: First Durarara!! original soundtrack, bundled with the fourth DVD
Release Date: 26 May 2010
Release Position: 4
His name takes the first half of the word "Psychedelic" from the album.
Official Name: Gakuen Tengoku (Gakuen Heaven)
Fanon Name: Gakuen
Source: Mikado's character song, bundled with the fifth DVD
Release Date: 23 June 2010
Release Position: 5
Official Name: Tsuppari High School Rock'n Roll
Fanon Name: Tsuppari
Source: Kadota's character song, bundled with the sixth DVD
Release Date: 21 July 2010
Release Position: 6

Official Name: Miraiegou (Forever)
Fanon Name: Miraiegou
Source: Second Durarara!! drama CD, bundled with the seventh DVD
Release Date: 25 August 2010
Release Position: 7
Official Name: Toki wo Kakeru Shoujo (The Girl Who Leapt Through Time)
Fanon Name: Toki
Source: Anri's character song, bundled with the eighth DVD
Release Date: 22 September 2010
Release Position: 8
Official Name: Psychedelic Dreams, vol. 02
Fanon Name: Delic
Source: Second Durarara!! original soundtrack, bundled with the ninth DVD
Release Date: 27 October 2010
Release Position: 9
His name takes the second half of the word "Psychedelic" from the album.
Official Name: Diamonds
Fanon Name: Diamonds
Source: Shinra's character song, bundled with the tenth DVD
Release Date: 24 November 2010
Release Position: 10
Official Name: Subarashii Hibi (Wonderful Days)
Fanon Name: Hibiya
Source: Izaya's character song, bundled with the eleventh DVD
Release Date: 22 December 2010
Release Position: 11
His name is a combination of "Hibi" and "Izaya".

Official Name: Rouge no Dengon (Lipstick Message)
Fanon Name: Rouge
Source: Celty's character song, bundled with the twelfth DVD
Release Date: 26 January 2011
Release Position: 12

Official Name: Tsukishima
Fanon Name: Tsuki
Source: Durarara!! Special DVD
Release Date: 23 February 2011
Release Position: 13
Tsukishima is actually the name of an area in Japan. The kanji on the pole denotes that the character is in the area but it also lends the character its name.

Official Name: N/A
Fanon Name: Shitsuo
Source: Kuroshitsuji crossover event
Release Date: N/A
Release Position: 14
Shizuo's voice actor, Ono Daisuke, also voiced Sebastian of Kuroshitsuji. There was a crossover event where Shizuo appeared in Sebastian's clothing and vice vers. Shitsuji, meaning "butler", was combined with Shizuo's name to create the name Shitsuo.

Official Name: N/A
Fanon Name: Sakuraya
Source: Yasuda artbook
Release Date: 30 June 2011
Release Position: 15
His name is a combination of "Sakura" and "Izaya".
All of the Durarara!! OSTs and characters songs can be downloaded here. The first drama CD can be downloaded here and the second drama CD can be downloaded here.
LATER APPEARANCES
Psyche, Delic, and Tsugaru were used as the mascots for a line of perfumes though I'm not sure this is official merchandise or not. Hand towels of the designs were given as freebies at the event if you bought them there.
Psyche and Delic were featured on a pillow which, as a user on Tumblr points out, the blue on the background of Psyche's image may suggest Tsguaru while the gold background on Delic's might suggest Hibiya. The pair is also featured together on a wallscroll among other merchandise.
The official art of the alternates appear in the Yasuda artbook along with other official images of the Durarara!! cast. Anyway, this is just to point out that these are not one time occurring designs! They've been integrated into the fandom, either as the original characters wearing different clothing or as alternate personalities for the fanbase.
OVERVIEW
So in terms of alternates, I thought I would explain some conventions of the fans.
Birthdays are the release dates! So, for example, Psyche's birthday is 26 May because the first Durarara!! OST was released on that date. I've included the release date along with the DVD the merchandise was bundled with for easy finding.
For the character songs, these are covers of songs. You might want to explore why the songs are relevant to the particular characters!
It's common in fanart that the nails of the alternates be painted in the color represented in the official art (pink for Psyche and Delic, red for Hachimenroppi and Tsukishima, etc.). This is not always the case but that's why, in case you were wondering! Whether this color is painted or natural is usually not addressed and can go either way.
The personality is derived from a fusion of the source material, the character design, and the original character. For example, Hibiya is depicted as a prince on a horse so he is thought of to be royalty or at least have a royal ego. His source material, Izaya's character song Subarashii Hibi, suggests that he doesn't want others to forget him; suggesting that he has abandonment issues might be valid headcanon. As an alternate Izaya, it might be inferred that he wants to be loved by all his subjects in a similar fashion that Izaya wants all humans to love him back.
In general, the alternates represent a part of the original's personality. This can be a wish (Tsugaru as Shizuo's desire to live a chillax, normal life), an opposite trait (Hachimenroppi as a misanthrope compared to Izaya's I♥HUMANS motto), an exaggerated trait (Delic as Shizuo's impulse and therefore exhausting his sex drive), or any reasonable trait you can justify. It is likely that the characters at least have some connection with their original because they share familiar traits: all the Izaya alternates are shown with mischievous expressions and all the Shizuo alternates are smoking (Tsukishima has a ticket in his mouth instead of a smoke, though).
PERSONALITIES
Okay, from here on out, we're going with just Shizaya alternates! I've decided to do this simply because there is a larger consensus in the fandom of what their personalities are. The personalities of the other characters are very vague and nobody will give you the same answer if you ask, whereas there's at least some basic grounding with the Shizaya alternates. Keep in mind that all the personalities can be fudged around with so there's no "true" interpretation.
Before we proceed: in the case of all the Shizuo alternates, there is nothing to say that all have or lack Shizuo's strength. Sometimes they're shown with Shizuo's strength and sometimes they're not. Both interpretations are technically correct.
Tsguaru: Tsugaru usually depicts Shizuo's desire to be calm and collected, the opposite of Shizuo. He smokes out a kiseru and is drawn in a haori. Combined with the fact he comes from Shizuo's character song which is an enka song, he is often depicted favoring the traditional.
Psyche: Depicted in completely white, Psyche is usually presented as the opposite personality of Izaya which can range from bubbly sweet to crybaby to mischievous. He is sometimes shown as a yandere.
Delic: It's pretty much agreed that Delic is an explosion of sex and in most situations, he's depicted as a host. Whether or not he's any good at this is disputed. Sometimes he's depicted using cheesy pickup lines and other times he's shown doing things that would make a porn star blush.
Hibiya: Hibiya is shown with a crown but it's not decided if he's a prince, if he has a kingdom, of if he's just delusional. Hibiya is depicted as a prude in a lot of situations as a foil for Delic. He is sometimes shown as a tsundere. Due the the nature of the material he's based off, Subarashii Hibi by UNICORN, many times he's shown to be lonely or afraid of people leaving him.
Tsukishima: Tsukishima is depicted as easily getting lost, shy, and messing up on simple things.
Hachimenroppi: He's interpretted as a misanthrope and sometimes suicidal. He often depicted with bandages from wounds he inflicted on himself. Because of the very delicate subjects surrounding Roppi, there's not much agreed about him.
Shitsuo: He is a recent addition to the fandom though it is mainly agreed that as a butler, he should be polite and carry out domestic duties. It would seem that he is based more on Sebastian from Kuroshitsuji more than Shizuo. He appears quiet and calm though some fanfic authors choose to display his internal struggle with his rage.
Sakuraya: He is the newest of the alternates and thus the least developed in terms of fandom consensus. So far he is depicted as a companion to Tsugaru and, like Tsugaru, is quiet, polite, and demure. He is sometimes demonstrated to rival Psyche for Tsugaru's affections.
PAIRINGS
I'm not getting into pairing wars so I'm just going to explain which are the ones that get paired up and why that might be. Generally, the Shizaya alternates break down into four pairings: Tsugaru/Psyche, Delic/Hibiya, Tsukishima/Hachimenroppi, and Shitsuo/Sakuraya.
Now if you're like me, you may have asked "well, why don't the pairings match up when the outfits obviously match?" Sure, the alts can be rearranged to match thematically as follows: Tsugaru/Sakuraya, Delic/Psyche, Tsukishima/Hachimenroppi, and Shitsuo/Hibiya. These are popular pairings in their own right, to be completely frank, though the mismatched pairings remain the more popular.
At first I thought that maybe the alternates were simply arranged in sequential order. Early fanart with the alternates only includes Psyche. Later Tsugaru is added so it's logical to pair them up. With Delic, trio fanart emerges. Some Psyche/Delic begins to flirt around the internet but the fanbase seems pretty loyal to Tsugaru/Psyche. When Hibiya rolls around, Delic is instantly paired with him and Tsugaru/Hibiya is virtually impossible to find. Tsukishima/Hachimenroppi and Shitsuo/Sakuraya are the logical followings (though Shitsuo/Hibiya and Tsugaru/Sakuraya have been depicted not so infrequently). So the passage of time may very well be the case. However, Hachimenroppi is technically Izaya's first alternate. Why Psyche caught on when Hachimenroppi didn't isn't something I can explain. However if Hachimenroppi's popularity is delayed until the appearance of Tsukishima, it explains why he didn't get paired up with an earlier Shizuo alternate. There's also the fact that Psyche gets far more merchandise than Hibiya, who in turn gets more merchandise than Hachimenroppi; so it may also be a matter of lack of exposure to the fans.
Basically the main quartet of pairings are popular, I'd wager, because they retain some form of the Shizaya hate-love complex and are based more on personality than looks. The following is the most objective headcanon I could form for each pairing but it is headcanon that I came up with. These are not the opinions of the entire fandom. If you agree with it, that's great, and if you don't, that's okay too! There's not one way these relationships can go. My aim is to explain why we have the general stereotype we have, regardless if we recognize the reasons behind these stereotypes.
Tsugaru/Psyche: These two are considered to be the opposite of Shizuo and Izaya and as such, presumably love each other on first sight rather than hate each other. Or, if you prefer to recognize them within the sphere of the Shizaya hate-sex relationship, Tsugaru and Psyche can be considered to be the personified relationship of Shizuo's and Izaya's complexes for each other (Shizuo hates only Izaya, Izaya hates only Shizuo, and yet a world without the other seems implausible). Tsugaru and Psyche are usually depicted as the understanding, stupidly sweet, idiotic couple.
Delic/Hibiya: These two are the quintessential hate-love aspect of the Shizaya relationship. Their relationship is characterized in a BDSM way in some cases, notably with Hibiya as the S and Delic as the M. Hibiya is often times depicted as a tsundere, refusing or rejecting Delic's playboy advances.
Tsuki/Roppi: These two are the hate-love aspect of Shizaya in reverse: whereas Roppi hates all humans, Tsukishima is the one person he doesn't hate. They are a subtle, serious couple that don't talk much compared to the first two. This pairing doesn't receive as much attention as the first two and is often problematic so there's not an agreed upon mode for their relationship.
Shitsuo/Sakuraya: Because these two are new, there's not yet much consensus on what their relationship is like. Thus far it seems that their relationship is a domestic one focused on supporting each other, especially through household chores.
FINAL NOTES
It depends heavily on your headcanon because they can be data programs or actual people. Even if they're decided to be people, their relationship with Shizuo and Izaya and then to each other is really complicated (cousins, brothers, lovers, etc.).
LINKS
A quick rundown of the different fan personalities and official art
Quick explanation of the personalities. Very condensed and precise!