The first view of the symbol is a photo taken by police at the site where a skeleton was found, and near where Kyle first awakes. Detective Breen shows the photo to Nicole along with others of the skeleton. Nicole shows it to Kyle as a possible way to trigger more memories, without showing him the skeleton. The back of the card is later shown to include the mysterious number 781227 along with a barcode that police files cannot match to any registered company or organization.
In this orientation, there seem to be two Z's with one rotated to cross the other like an X, suggesting Zzyzx. However, no explanations are given of the symbol or the name.
Kyle has been making many drawings, in his unique pointillism pixel style, of the place in the woods where he awoke. Nicole is putting them on a bulletin board in Kyle's new room, and happens to see the odd, irrelevant black parts as fitting together like a puzzle. She is shocked, and troubled, to see that they make the symbol associated with the murder.
When Tom Foss resigns from "the company", he of course must give back his security key card. From that, we know that the other card was associated with the same "company", and that Foss may have been involved with the murder of William Kern.
When Declan and Lori take Kyle with them to the Univ. of Washington, Kyle finds an exciting sketch sheet drawn by Adam Baylin, the student in the 1985 photo who looked exactly like Kyle. That connection makes the map grid coordinates on the same sheet into a quest for Kyle and Declan.
When Kyle is living with Adam in season 2, Adam shows him a page in his study journals with further sketches of the symbol.
Rebecca Thatcher as Director at Zzyzx is determined to have 781227 terminated, now to prevent any news of the experiment being known by way of Kyle. On her desk is a fishbowl decorated with an etching of the Zzyzx symbol. The symbol is also painted on doors inside the underground complex. But the next person to see anything of the symbol, is 781228.
When 781228 is inside the shipping container after being liberated, she has managed to make a lot of drawings in the dark, but none contain the symbol as Kyle's did. Inside the Zzyzx compound on the fateful outing in Ghost in the Machine, Jessi is trapped in the room where her gestation tank had been, and she is also trapped in her first memories, which are of the fire-bombing. She sees the symbol on the wall within the flames.
Later, during a therapy session, Nicole suggests that she draw something. It is of the flames. Nicole later thinks she can see the symbol within the patterns of fire. She adds it herself, and adds to her notion that Jessi is too much like Kyle for some connection to be ignored.
We do not see the symbol again.