  We consider time reversal transformations to obtain twofold orthogonal
splittings of any tensor on a Lorentzian space of arbitrary dimension n.
Applied to the Weyl tensor of a spacetime, this leads to a definition of its
electric and magnetic parts relative to an observer (i.e., a unit timelike
vector field u), in any n. We study the cases where one of these parts vanishes
in particular, i.e., purely electric (PE) or magnetic (PM) spacetimes. We
generalize several results from four to higher dimensions and discuss new
features of higher dimensions. We prove that the only permitted Weyl types are
G, I_i and D, and discuss the possible relation of u with the WANDs; we provide
invariant conditions that characterize PE/PM spacetimes, such as Bel-Debever
criteria, or constraints on scalar invariants, and connect the PE/PM parts to
the kinematic quantities of u; we present conditions under which direct product
spacetimes (and certain warps) are PE/PM, which enables us to construct
explicit examples. In particular, it is also shown that all static spacetimes
are necessarily PE, while stationary spacetimes (e.g., spinning black holes)
are in general neither PE nor PM. Ample classes of PE spacetimes exist, but PM
solutions are elusive, and we prove that PM Einstein spacetimes of type D do
not exist, for any n. Finally, we derive corresponding results for the
electric/magnetic parts of the Riemann tensor. This also leads to first
examples of PM spacetimes in higher dimensions. We also note in passing that
PE/PM Weyl tensors provide examples of minimal tensors, and we make the
connection hereof with the recently proved alignment theorem. This in turn
sheds new light on classification of the Weyl tensors based on null alignment,
providing a further invariant characterization that distinguishes the types
G/I/D from the types II/III/N.
