  We describe conventional orientifold and orbifold quotients of string and
M-theory in a unified approach based on exceptional field theory (ExFT). Using
an extended spacetime, ExFT combines all the maximal ten and eleven dimensional
supergravities into a single theory manifesting a global symmetry corresponding
to the exceptional series of Lie groups. Here we will see how this extends to
half-maximal theories by showing how a single $\mathbb{Z}_2$ generalised
orbifold (or O-fold) of ExFT gives rise to M-theory on an interval, type II
with orientifold planes and the heterotic theories in an elegant fashion. We
study in more detail such orbifold and orientifold actions preserving
half-maximal supersymmetry, and show how the half-maximal structure of ExFT
permits the inclusion of localised non-Abelian vector multiplets located at the
orbifold fixed points. This allows us to reproduce for the $\mathbb{Z}_2$
example the expected modifications to the gauge transformations, Bianchi
identities and actions of the theories obtained from the single ExFT starting
point. We comment on the prospects of studying anomaly cancellation and more
complicated, non-perturbative O-folds in the ExFT framework.
