We study solutions in the Plebański–Demiański 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 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,
W
C
P
1
[
n
−
,
n
+
]
, 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
AdS
2
×
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.