  The notion of symmetric and asymmetric peaks in Dyck paths was introduced by
Fl\'orez and Rodr\'{\i}guez, who counted the total number of such peaks over
all Dyck paths of a given length. In this paper we generalize their results by
giving multivariate generating functions that keep track of the number of
symmetric peaks and the number of asymmetric peaks, as well as the widths of
these peaks. We recover a formula of Denise and Simion as a special case of our
results.
  We also consider the analogous but more intricate notion of symmetric
valleys. We find a continued fraction expression for the generating function of
Dyck paths with respect to the number of symmetric valleys and the sum of their
widths, which provides an unexpected connection between symmetric valleys and
statistics on ordered rooted trees. Finally, we enumerate Dyck paths whose peak
or valley heights satisfy certain monotonicity and unimodality conditions,
using a common framework to recover some known results, and relating our
questions to the enumeration of certain classes of column-convex polyominoes.
