  The congruential rule advanced by Graves for polarization basis
transformation of the radar backscatter matrix is now often misinterpreted as
an example of consimilarity transformation. However, consimilarity
transformations imply a physically unrealistic antilinear time-reversal
operation. This is just one of the approaches found in literature to the
description of transformations where the role of conjugation has been
misunderstood. In this paper, the different approaches are examined in
particular in respect to the role of conjugation. In order to justify and
correctly derive the congruential rule for polarization basis transformation
and properly place the role of conjugation, the origin of the problem is traced
back to the derivation of the antenna hight from the transmitted field. In
fact, careful consideration of the role played by the Green's dyadic operator
relating the antenna height to the transmitted field shows that, under general
unitary basis transformation, it is not justified to assume a scalar
relationship between them. Invariance of the voltage equation shows that
antenna states and wave states must in fact lie in dual spaces, a distinction
not captured in conventional Jones vector formalism. Introducing spinor
formalism, and with the use of an alternate spin frame for the transmitted
field a mathematically consistent implementation of the directional wave
formalism is obtained. Examples are given comparing the wider generality of the
congruential rule in both active and passive transformations with the
consimilarity rule.
