Why, intuitively, is the order reversed when taking the transpose of the product? It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$
and a nice way for me to remember this is the following sentence:


  The opposite of putting on socks and shoes is taking the shoes off, followed by taking the socks off.


Now, a similar law holds for the transpose, namely:

$$(AB)^T=B^TA^T $$

for matrices $A,B$ such that the product $AB$ is defined. My question is: is there any intuitive reason as to why the order of the factors is reversed in this case? 

[Note that I'm aware of several proofs of this equality, and a proof is not what I'm after]

Thank you!
One of my best college math professor always said:


  Make a drawing first.




Although, he couldn't have made this one on the blackboard.
My professor always called them the shoe-sock theorems or the likes, the idea basically being that you put on your socks first then the shoes, but to do the reverse you must get off the shoes first and then the socks.