  The two-sided quaternionic Fourier transformation (QFT) was introduced in
\cite{Ell:1993} for the analysis of 2D linear time-invariant
partial-differential systems. In further theoretical investigations
\cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions
was introduced, then called $\pm$split. In the current \change{chapter} we
analyze this split further, interpret it geometrically as \change{an}
\emph{orthogonal 2D planes split} (OPS), and generalize it to a freely
steerable split of $\H$ into two orthogonal 2D analysis planes. The new general
form of the OPS split allows us to find new geometric interpretations for the
action of the QFT on the signal. The second major result of this work is a
variety of \emph{new steerable forms} of the QFT, their geometric
interpretation, and for each form\change{,} OPS split theorems, which allow
fast and efficient numerical implementation with standard FFT software.
