This paper deals with one-dimensional bidirectional sequences (a) under bar : Z --> V, V a finite set, such that any p-decimation (\p \ greater than or equal to 2) of the sequence reproduces the sequence (modulo a certain shift). We develop a procedure for solving the underlying decimation-invariance (DI) equations and find that the number of solutions is always finite. Conditions for equivalency among solutions of differently parametrized DI-problems, and for possible periodicity and symmetry of solutions, are derived. It is shown that the set of all possible p-based decimations of a such a DI sequence (the so-called full kernel of the sequence) is finite. This implies finiteness of the kernel for the separate right and left parts of the sequence, and also \p \ -automaticity of these parts. An algorithm is presented that constructs the kernel and associated \p \ -automaton of a DI-sequence explicitly. (C) 2001 Elsevier Science B.V. All rights reserved.