The third line of the show's theme song nearly 100% references CatDog's adoptive parents.
When the Greasers mark their territory on the fake golden hydrant in "The Golden Hydrant", Shriek is seen lifting her leg even though she is a girl. Considering her tomboyish nature, though, it actually would make sense that she'd hike her leg rather than crouch.
Besides, in Real Life, female dogs sometimes do lift their legs, especially when marking their territory.
In "The Island", Dog says he wishes Winslow was with them when they are suffering. In "The Great Parent Mystery", it is revealed that Winslow was the one who raised CatDog from the time they were babies, so it can be likened to a child saying I Want My Mommy!
If Winslow keeps torturing CatDog, and he's a freeloader, why don't they kick him out of the house? When they were babies and had no home, Winslow took them in and raised them himself. It's likely that the twins are returning the favor.
In "Back to School", a flashback shows a teenaged Cat unable to do a rope-climbing exercise in gym class. Of course he couldn't do it; not only is he physically weak, he has a conjoined twin attached to him, who would likely be difficult to drag along with him.
Cat and Dog are named simply that. Cat and Dog. It makes sense when you remember that Winslow was the one who found them. Of course he wouldn't care enough to give them names.
For the large part, Dog gets off the easiest throughout the show, supposedly because he is too dumb to know better. One of the clear exceptions to this rule is "Smarter Than The Average Dog" where Cat is allowed to pummel the crap out of Dog for stealing all their brain matter. Why? Because Dog wasn't dumb in that case, meaning he had no clause for being inconsiderate anymore.
CatDog's very nature as a pair of Conjoined Twins consisting of A Head at Each End raises a lot of disturbing Body Horror-related questions about how their body works, namely how they go to the bathroom.
It is established in "Nine Lives" that Cat has nine lives. Given that he and Dog share a body, one can't help but wonder what would happen if Dog outlived all of Cat's lives, or for that matter, if Cat outlived Dog. There are two possibilities, both of which aren't too good: The first possibility: they both would die since they share the same body, dying much too young and much too soon. The second possibility is even worse: if one died and the other continued to live, their other half would be the corpse of their brother, a sure-fire hell of a reminder that the one that survived would be alone for the rest of their life, having just lost the companion they've known since birth. Their literal other half, whether in soul or in body, would be utterly gone.
If one died, the other would die too. Think of it like "If I'm going down, your going down with me."
In "Remain Seated", Cat imagines his own death. His half is buried while Dog's remains above ground. This also evokes Fridge Horror: either Dog, who is still alive, can't move away from the grave, or if he does try to move, he brings his brother's buried corpse to the ground. Or worse, like in the "Scientifically Accurate" parody, necrosis would set in, and Dog would die too as a result of being attached to a corpse.
There might be a chance to avoid double-death if they can be surgically separated (or at least the deceased head and affected body parts removed) immediately after Cat or Dog dies, but that will also depend on the biological makeup of their body. Though, even if this is possible it would still be quite the traumatic procedure both in the physical sense and as well as what it means for the survivor.
Near the end of "Meat Dog's Friends", Dog finds himself unable to eat anything he's made friends with, so he tries to eat his conjoined brother. What could have happened if Johnny Meatseed didn't show up to plant meat trees for Dog so he'd be able to eat without worrying about eating dead animals?
To try and complete Dog's initiation as one of the Greasers in "Full Moon Fever", the Greasers prepare to glue a tail to Cat's head, which Shriek claims is genuine Dalmatian. Nothing at all is said of whose tail it was or how the Greasers obtained it, which gives the impression that they may have lopped off an innocent Dalmatian's tail.
The Halloween Episode "CatDogula" ends with Cat curing everyone, including the Peruvian vampire ticks, of their vampirism. Since the Peruvian vampire ticks were also cured rather than killed, this could mean that they were victims of another vampire that is still at large.
In "CatDog Doesn't Live Here Anymore", the Greasers ask Winslow "Hey, can CatDog come out and run away from us, please?" like a child asking a friend's parent if the friend can come out and play. The Fridge Horror here kicks in when you realize that Winslow adopted the brothers and this implies that all these years, even when they were kids, Winslow, their Parental Substitute, agreed to let the Greasers beat up CatDog.
