  We study solutions in the Pleba\'nski--Demia\'nski family which describe an
accelerating, rotating and dyonically charged black hole in $AdS_4$. These are
solutions of $D=4$ Einstein-Maxwell theory with a negative cosmological
constant and hence minimal $D=4$ gauged supergravity. It is well known that
when the acceleration is non-vanishing the $D=4$ black hole metrics have
conical singularities. By uplifting the solutions to $D=11$ supergravity using
a regular Sasaki-Einstein $7$-manifold, $SE_7$, we show how the free parameters
can be chosen to eliminate the conical singularities. Topologically, the $D=11$
solutions incorporate an $SE_7$ fibration over a two-dimensional weighted
projective space, $\mathbb{WCP}^1_{[n_-,n_+]}$, also known as a spindle, which
is labelled by two integers that determine the conical singularities of the
$D=4$ metrics. We also discuss the supersymmetric and extremal limit and show
that the near horizon limit gives rise to a new family of regular
supersymmetric $AdS_2\times Y_9$ solutions of $D=11$ supergravity, which
generalise a known family by the addition of a rotation parameter. We calculate
the entropy of these black holes and argue that it should be possible to derive
this from certain ${\cal N}=2$, $d=3$ quiver gauge theories compactified on a
spinning spindle with appropriate magnetic flux.
