We study the phenomena produced when inversion of direction is allowed when reading a word. Two-way reading on a given language x can be reduced to left-to-right reading on a language containing x, which is regular whenever x is so. We present two characterizations of "zig-zag languages". We then consider and compare two possible ways of counting two-way reading on a regular language, and thus, of defining the behaviour of two-way automata. For each one definition, we show the construction of a one-way automaton equivalent in multiplicity to a given two-way automaton, this generalizing Rabin, Scott and Shepherdson's Theorem.
