What it says on the tin. The HTML center tag doesn't work, I know that at least historically, div align was blocked, and I can't find where text alignment is in the new text editor.
Thanks @Desnik; going out of HTML mode and back into the text editor worked. That's...kind of stupid, I've gotta say. Not sure what the point is of having a code editor if you need to go back into the rich text if you want any edits to actually stick. Apparently you can't actually save anything and have it stick if you're editing in code mode without going back to WYSIWYG. Here's my video of the behavior.