I honestly haven't thought about it even once since the first episode, up to only seeing friskuser's 4koma here today.
Quite funny, the golden animal head on her military hat really is something of a face, so someone can see it as Altair's "true demeanor". That's just mocking her though. The headgear is just a part of her dress-up. She isn't less of herself without her accessories in the same way Magane is still Magane without her earrings, Meteora without her roundish white-blue hair ornaments is still Meteora or even Hatsune Miku without her twintails is still Hatsune Miku - otaku's beloved diva. In my opinion, even if Sirius hadn't worn her hat, Altair would still have overthrown her identity, as she already was doing so to make her wear that hat.

Nozek said:

I honestly haven't thought about it even once since the first episode, up to only seeing friskuser's 4koma here today.
Quite funny, the golden animal head on her military hat really is something of a face, so someone can see it as Altair's "true demeanor". That's just mocking her though. The headgear is just a part of her dress-up. She isn't less of herself without her accessories in the same way Magane is still Magane without her earrings, Meteora without her roundish white-blue hair ornaments is still Meteora or even Hatsune Miku without her twintails is still Hatsune Miku - otaku's beloved diva. In my opinion, even if Sirius hadn't worn her hat, Altair would still have overthrown her identity, as she already was doing so to make her wear that hat.

Actually, as I see it, Altair took over even before Sirius wore that hat.
But to those who weren't looking, it's possible for them to see it that way. Just like Kujou Sakurako said in "Sakurako-san no Ashimoto ni Shitai ga Umatteiru" series, they can really see it that way.

But then, that's only a joke. Like Touhou's characters are not themselves anymore without their head gears, be it a hat or a ribbon.

Oh it's platyp-PERRY THE PLATYPUS?!?