  It has been established that Christodoulou's formulation of Strong Cosmic
Censorship (SCC) is violated by Reissner-Nordstr\"om-de Sitter black holes, but
holds in four-dimensional Kerr-de Sitter black holes. We show that SCC is also
respected by equal angular momenta (cohomogeneity-1) Myers-Perry-de Sitter
(MP-dS) in odd $d \ge 5$ spacetime dimensions. This suggests that the
preservation of SCC in rotating backgrounds might be a universal property of
Einstein gravity and not limited to the d = 4 Kerr-dS background. As required
to discuss SCC in de Sitter spacetimes, we also study important aspects of the
scalar field quasinormal mode (QNM) spectra of MP-dS. In particular, we find
eigenvalue repulsions similar to those recently observed in the QNM spectra of
asymptotically flat Kerr-Newman black holes. For axisymmetric modes (i.e. with
azimuthal quantum number m = 0) there are three distinct families of QNM (de
Sitter, photon sphere and near-horizon). However, typically, for
non-axisymmetric ($m \ne 0$) QNMs, we find that the entire spectra can be
described by just two families of QNM (since several overtone sections of the
photon sphere and near-horizon families merge). For completeness, we also study
the full scalar field QNM spectra of higher-dimensional Schwarzschild-de Sitter
black holes.
