  We describe an alternative approach to the analysis of gravitational-wave
backgrounds, based on the formalism used to characterise the polarisation of
the cosmic microwave background. In contrast to standard analyses, this
approach makes no assumptions about the nature of the background and so has the
potential to reveal much more about the physical processes that generated it.
An arbitrary background can be decomposed into modes whose angular dependence
on the sky is given by gradients and curls of spherical harmonics. We derive
the pulsar timing overlap reduction functions for the individual modes, which
are given by simple combinations of spherical harmonics evaluated at the pulsar
locations. We show how these can be used to recover the components of an
arbitrary background, giving explicit results for both isotropic and
anisotropic uncorrelated backgrounds. We also find that the response of a
pulsar timing array to curl modes is identically zero, so half of the
gravitational-wave sky will never be observed using pulsar timing, no matter
how many pulsars are included in the array. An isotropic, unpolarised and
uncorrelated background can be accurately represented using only three modes,
and so a search of this type will be only slightly more complicated than the
standard cross-correlation search using the Hellings and Downs overlap
reduction function. However, by measuring the components of individual modes of
the background and checking for consistency with isotropy, this approach has
the potential to reveal much more information. Each individual mode on its own
describes a background that is correlated between different points on the sky.
A measurement of the components that indicates the presence of correlations in
the background on large angular scales would suggest startling new physics.
