My intuition of trace is that it is essentially asking, for each basis, where does this basis vector go, and what is the new vector's component in its original basis. Then, trace is a sum of those terms.
And matrix non-commutativity makes sense. You apply one transformation, then the next. And this can lead to differences in the result of $AB$ and $BA$.
But if matrices multiplication is non-commutative, then why, in light of my intuition of trace, is trace cyclic. $$\operatorname{trace}(AB)=\operatorname{trace}(BA)$$
I can show it numerically, but I'd like to be able to think of it in a more intuitive and geometric way.