So I had a little bit of fun. All that I meant by commutative inverses was this:

Let `b` be the operation of backspacing, & `e` be the operation of typing `=`. Clearly `b` is left-inverse for `e` as
> `e b = ''`
> `b e = e`
Suppose that `s` and `n` are the strict and nonstrict members of an given inequality. Then the action of `b` and `e` on `s` and `n` such that
> `n * b    = s` 
> `s * e    = n`
> `n * b e = n`
> `s * e b = s`

If, after directly typing `<`, typing `=` does the right thing, but then when you backspace, the resulting node correctly outputs LaTeX of `<` but its state is messed up someotherhow such that typing `=` again won't work, then typing `=` and backspace would not be inverse operations as they didn't preserve the state of `<`. 

Similarly if we start with `≤`, backspace and it does the right thing, but when you type `=`, the resulting node correctly outputs LaTeX of `\le` but its state is messed up such that backspace doesn't work correctly, then backspace and typing `=` would not be inverse operations as the state of `≤` was not preserved.