  We study the dimension theory of a class of planar self-affine multifractal
measures. These measures are the Bernoulli measures supported on box-like
self-affine sets, introduced by the author, which are the attractors of
iterated function systems consisting of contracting affine maps which take the
unit square to rectangles with sides parallel to the axes. This class contains
the self-affine measures recently considered by Feng and Wang as well as many
other measures. In particular, we allow the defining maps to have non-trivial
rotational and reflectional components. Assuming the rectangular open set
condition, we compute the $L^q$-spectrum by means of a $q$-modified singular
value function.
  A key application of our results is a \emph{closed form expression} for the
$L^q$-spectrum in the case where there are no mappings that switch the
coordinate axes. This is useful for computational purposes and also allows us
to prove differentiability of the $L^q$-spectrum at $q=1$ in the more difficult
`non-multiplicative' situation. This has applications concerning the Hausdorff,
packing and entropy dimension of the measure as well as the Hausdorff and
packing dimension of the support. Due to the possible inclusion of axis
reversing maps, we are led to extend some results of Peres and Solomyak on the
existence of $L^q$-spectrum of self-similar measures to the graph-directed
case.
