Why are winter jackets not commutative? Say I have two jackets $A$ and $B$, which have some fixed parameters such as thermal insulation. Further assume each jacket can be worn on top of the other one without changing its parameters. Let $A\circ B $ be the clothing situation which arises from putting on $B$, and then $A$, and let $B \circ A$ be the reverse.
In real life, $A \circ B$ and $B \circ A$ provide different amounts of warmth. For example in winter sports, people wear a warm inner layer and a windbreaking outer layer—presumably the reverse is not as effective.
What's the simplest thermodynamic model of "how jackets work" which explains this noncommutative behavior?
Really, what this question is asking is how to compute the warmth provided by $A \circ B$ and $B \circ A$ given some parameters of jackets $A,B$. For example, if the only information we were given was $p_A, p_B$, the percentage of body heat retained by $A$ and $B$, then my guesses for $p_{A \circ B}$ would be expressions like
$$ \min(1, p_A + p_B),\qquad 1-(1-p_A)(1-p_B),\qquad $$
and maybe also applying some sort of barrier function to make sure these estimates never actually hit $1$. However, all of these expressions are symmetric in $A$ and $B$, so they do not explain the noncommutative behavior of jackets.
 A: To paraphrase Tolstoy, all commutative thermal insulators in series are alike; noncommutative thermal insulators in series are different in their own way.
For example, insulators may have different effective convective coefficients (because of different roughness, for example). Open weaves may have different effective surface area or thickness (because cold air can convectively permeate them). The thermal conductivity of absorbent materials may be strongly affected by the absorbed moisture, whereas nonabsorbent materials are robust against moisture. The exterior color affects radiative heat transfer. The thermal conductivity varies with temperature. All these factors—among others—break symmetry and preclude simply adding thermal resistances together except in the simplest, most idealized model.
A: I think there are two important things.
(1) We have a human inside of jackets and an environment outside. The environtment can probably be modeled as a kind temperature bath/uniform system, but the human inside is complicated. In particular, the Human inside is converting chemical bonds into heat, they're also generating sweat. The human also has feedback systems in place where their dynamics change if their temperature changes. For example, they shiver if they get cold. So this complication of the human already takes away from a very simple thermal conductivity heat-flow model for jackets.
(2) This is probably the most important: Some of the insulating properties of certain thermal insulators fails if the insulator gets wet. For example a down or synthetic down jacket gets it's thermal resistance from air pockets maintained by the microscopic structure of the insulating materials. When the insulator gets wet the fibers collapse and thermal resistance drops dramatically.
The second point leads to the very simple conclusion that if it is raining and a rain jacket is worn under a rain jacket it will keep you much warmer than if a down jacket is worn over a rain jacket.
There are other similar considerations, for example, if you are not wearing a nice wicking base layer next to your skin then the sweat from skin will not be properly handled and this sweat can subsequently freeze and keep you cooler than if you had managed it with a different layering order.
In short... it's not just a simple 1D thermal conductivity problem AT ALL and there's no "one property" of the insulating jackets that breaks this "commutativity".
