Every so often, I'll see Catra stans complain that Hordak showed no remorse, with the intended point being that Catra deserved redemption, while he didn't, because she did show remorse.

And it's a sort of logical gut-reaction, but it doesn't match the character patterns.
Hordak doesn't express remorse because that's not really the point of his character arc. Well, and also because the show doesn't focus on him, so he's not as developed, but in terms of writing, he and Catra face different challenges during their arcs.
It's important to see Catra express remorse, however clumsily written, because the entire point of her character is her making terrible decisions, avoiding responsibility, and blaming others as a result of her own trauma. Her challenge is to recognize that she is in control-
-and is thus in part responsible for her circumstances. This is highlighted by Adora in season three and by Double Trouble in season four. It's her big point of character growth: recognizing that she's hurt everyone around her and owning it so she can have friends again.
Hordak, while certainly responsible for significant damage, has a different challenge.

Unlike Catra, he doesn't realize that anything is wrong. He's not aware of the fact that he's abused, so he's not really capable of blaming it on anyone.
He's not seeking to punish others for his pain. Rather, he sees himself as the problem and seeks to "fix" himself in the way his indoctrination dictates is necessary.

Hordak doesn't need to recognize that he's made bad choices. He needs to recognize that he *has* a choice.
He needs to recognize that what Prime is doing to him and demanding of him is wrong. He needs to recognize that *he* is not the problem: Prime is.
Catra already knows that Shadow Weaver is a problem. She starts off knowing this. What she needs to learn is that she, herself, needs to change. Her own behavior is toxic, and she spends the series learning and accepting this. This acceptance is then met with remorse.
Hordak simply starts off a step behind: he doesn't realize that Prime, essentially his Shadow Weaver, is a problem. And so *that* is his challenge for the series. And in order to meet it, he doesn't need to express remorse. That's not the point of his arc.
Now, will he eventually have to? Probably, if he intends to interact WITH Etherians in any appreciable capacity. But we don't see it on the show because it's not the time, nor the place, nor the point of his storyline.
tl;dr Catra has to express remorse for making bad choices. Hordak has to first realize that he has choices that he is able to make.
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