  Within five-dimensional compactified theories we discuss generalized
periodicity and orbifold boundary conditions that allow for mixing between
particles and anti-particles after a shift by the size of extra dimensions or
after the orbifold reflection. A systematic strategy for constructing
4-dimensional models is presented, in particular we find a general form of the
periodicity and orbifold conditions that are allowed by consistency
requirements. We formulate general conditions for a presence of massless
Kaluza-Klein modes and discuss remaining gauge symmetry of the zero-mode
sector. It is shown that if the orbifold twist operation transforms particles
into anti-particles then the zero-mode fermions are 4-dimensional Majorana
fermions. The possibility of explicit and spontaneous CP violation is
discussed. General considerations are illustrated by many Abelian and
non-Abelian examples.
