  In this paper we present a characterization of the symmetric rotational
periodic gravity water waves of finite depth and without stagnation points in
terms of the underlying flow. Namely, we show that such a wave is symmetric and
has a single crest and trough per period if and only if there exists a vertical
line within the fluid domain such that all the fluid particles located on that
line minimize there simultaneously their distance to the fluid bed as they move
about. Our analysis uses the moving plane method, sharp elliptic maximum
principles, and the principle of analytic continuation.
