Name for the fact that a mattress can't be evenly rotated by repeatedly applying the same transformation?

The rotations of the mattress can be represented by an algebraic structure called a group. This group in particular is called the Klein four-group and is isomorphic to the direct product of two copies of $\mathbb Z / 2\mathbb Z$. (One of those copies is the exchange (T,B), and the other is (N,S)). The group has the property that every operation (except for the trivial "do nothing" operation) has order $2$, i.e., any operation performed twice will bring the mattress to its original state.

If all the elements of a group can be attained by repeatedly applying one operation $g$, the group is said to be cyclic and can be generated by $g$. In the case of the mattress, since no operation has order 4, the group is not cyclic. It is generated by the two flips (T,B) and (N,S), but not by a single flip.

The result you are wondering about with the more general case of the direct product of $\mathbb Z / m\mathbb Z$ and $\mathbb Z / n\mathbb Z$ (i.e., "cycling through substates $m$ and $n$") is described in the Wikipedia article on the direct product of groups. In short, $\mathbb Z / m\mathbb Z \times \mathbb Z / n\mathbb Z$ is cyclic and generated by $(1,1)$ if and only if $\gcd(m,n) = 1$.

(thank you to @pjs36 for pointing out the missing critical terms!)

I can make a suggestion. Let us call this season Spring. So write the word Spring in permanent marker under where your head goes. In three months, put some other edge in that position, and write the word Summer there. Three months later, one of the edges not used yet, and write Autumn. Three months later, Winter.

After that, every three months, search for the corner that has the upcoming season.

If you move between the North and South hemispheres, buy a new mattress.

If you don't want to use marker, maybe embroidery. Cross-Stitch.