  The transverse field structure and diffraction loss of the resonant modes of
Fabry-P\'erot optical cavities are acutely sensitive to the alignment and shape
of the mirror substrates. We develop extensions to the `mode mixing' method
applicable to arbitrary mirror shapes, which both facilitate fast calculation
of the modes of cavities with transversely misaligned mirrors and enable the
determination and transformation of the geometric properties of these modes. We
show how these methods extend previous capabilities by including the
practically-motivated case of transverse mirror misalignment, unveiling rich
and complex structure of the resonant modes.
