  In 4 spacetime dimensions there is a well known proof that for any
asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein's
equation there exists a "$t$-$\phi$" reflection isometry that reverses the
direction of the timelike Killing vector field and the direction of the axial
Killing vector field. However, this proof does not generalize to higher
spacetime dimensions. Here we consider asymptotically flat, stationary, and
axisymmetric (i.e., having one or more commuting rotational isometries) black
hole spacetimes in vacuum general relativity in $d \geq 4$ spacetime dimensions
such that the action of the isometry group is trivial. (Here "trivial" means
that if the "axes"---i.e., the points where the axial Killing fields are
linearly dependent---are removed, the action of the isometry group is that of a
trivial principal fiber bundle. This excludes actions like that found in the
Sorkin monopole.) We prove that there exists a "$t$-$\phi$" reflection isometry
that reverses the direction of the timelike Killing vector field and the
direction of each axial Killing vector field. The proof relies in an essential
way on the first law of black hole mechanics.
