Abstract
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.
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