We study solutions in the Plebański–Demiański family which describe an accelerating, rotating, and dyonically charged black hole in ${\mathrm{AdS}}_4$. 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 nonvanishing the $D=4$ black hole metrics have conical singularities. By uplifting the solutions to $D=11$ supergravity using a regular Sasaki-Einstein seven-manifold, $S{E}_7$, we show how the free parameters can be chosen to eliminate the conical singularities. Topologically, the $D=11$ solutions incorporate an $S{E}_7$ fibration over a two-dimensional weighted projective space, $WC{P}_{[n_{-},n_{+}]}^1$, also known as a spindle, which is labeled 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 ${\mathrm{AdS}}_2\times Y_9$ solutions of $D=11$ supergravity, which generalize 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 $N=2$, $d=3$ quiver gauge theories compactified on a spinning spindle with the appropriate magnetic flux.