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The effective reluctance between the two windings should remain same, no matter on which arms they are placed. The pair of arms with less cross sectional area will offer more reluctance as compared to the other pair. Since in both cases, flux passes through both pair of arms so effective reluctance should remain same in both cases and hence the mutual Inductance. Whats wrong with this explanation?
The mutual inductance can be found as $$L=\frac { N_1 N_2 } \Re$$ where \$N_1\$ and \$N_2\$ is the number of turns of each winding and \$\Re\$ is the reluctance of the magnetic closed circuit.
The reluctance is independent of the position of the windings, it just depends on the geometry of the core (and the core material). Therefore the mutual inductance remains the same.
Reluctance of a transformer core is: -
\$\dfrac{\text{effective length of core}}{\mu\times \text{effective area of core}}\$
The effective length is the mean length thru the core from beginning to end and, of course, this is the same starting at any point. The effective area is the "average" cross sectional area of ALL the iron and this is constant for both coil configurations.
No matter where you place the coils (within reason) and within a few percent (not 50%) the mutual coupling will be the same.
