Definitions

First some definition must be done:

Reciprocal Networks:
Two networks $A$ and $B$ are reciprocal to each other if their transimpedances have the following relation:

$$Z_{mn,A} = Z_{nm,B}$$ (5.1)

That means: Drive the current $I$ into node $n$ of circuit $A$ and at node $m$ the voltage $I\cdot Z_{mn,A}$ appears. In circuit $B$ it is just the way around.

Adjoint Networks:
Network $A$ and network $B$ are adjoint to each other if the following equation holds for their MNA matrices:

$$[A]^T = [B]$$ (5.2)