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There can be plenty of heat exchange in and out of all kinds of surfaces with "Adiabatic" boundary condition regardless of symmetry in the construction. The same boundary conditions get applied to both faces of the surface, but the surface still can have heat capacity and it is likely to be hotter or colder than the surroundings at times. An asymmetrical construction might not end up with symmetrical surface temperature results at the faces at any given time, but over time it will average out. It is just a temporal artifact of how thermal storage charging and discharging happens.