  For sampling values along spherical Lissajous curves we establish a spectral
interpolation and quadrature scheme on the sphere. We provide a mathematical
analysis of spherical Lissajous curves and study the characteristic properties
of their intersection points. Based on a discrete orthogonality structure we
are able to prove the unisolvence of the interpolation problem. As basis
functions for the interpolation space we use a parity-modified double Fourier
basis on the sphere which allows us to implement the interpolation scheme in an
efficient way. We further show that the numerical condition number of the
interpolation scheme displays a logarithmic growth.
  As an application, we use the developed interpolation algorithm to estimate
the rotation of an object based on measurements at the spherical Lissajous
nodes.
