  Interference is widely regarded as a foundational attribute of quantum
mechanics. However, for a given experimental arrangement, interference can
either contribute or not contribute to the outcome depending upon the basis in
which it is measured. This observation is both foundational and particularly
relevant to coherent control of molecular processes, an approach based upon
quantum interference. Here we address this issue and its relevance to
controlling molecular processes via the "coherent control scattering (CCS)
matrix", a formalism that allows an analysis of modifications in interference
structure resulting from a change of basis. This analysis reveals that the
change in interference structure can be attributed to the non-commutativity of
the transformation matrix with the CCS matrix, and the non-orthogonality of the
transformation. Additionally, minimal interference is shown to be associated
with the CCS eigenbasis, and that the Fourier transform of the eigenvectors of
the CCS matrix provides the maximal interference and hence the best coherent
control. The change of controllability through a change of basis is illustrated
with an example of $^{85}$Rb+ $^{85}$Rb scattering. In addition, the developed
formalism is applied to explain recent experimental results on He + D$_2$
inelastic scattering demonstrating the presence or absence of interference
depending on the basis.
