  We consider quasiradial Fourier multipliers, i.e. multipliers of the form
$m(a(\xi))$ for a class of distance functions $a$. We give a necessary and
sufficient condition for the multiplier transformations to be bounded on $L^p$
for a certain range of $p$. In addition, when $m$ is compactly supported in
$(0,\infty)$, we give a similar result for associated maximal operators.
