If a transformer's primary and secondary are connected in series, then the total inductance of the result, is greater than the summation of each coil separately.

That is to say: \$ L_{total} > (L_p + L_s) \$.
I've seen the math for this and understand it well enough; but the thing is, if I connect an xfmrs coils in series and then short the primary (so that there is a direct path to the secondary) the result is significantly greater than the inductance of the secondary by itself: this I don't understand.
I understand that the short is not zero (unless one is using a super conductor), and that there is going to be current through the primary, but there seems to be more to it. For instance, if I short the secondary (instead of the primary), the LCR meter reads MUCH higher than as if the coils were just connected in series!
I want to make sure that my LCR meter is correct, because if this phenomenon holds true, then I would like to take advantage of this in a filter circuit.
Can someone help me to understand what's going on?
