  A unified approach to embedding theorems for Sobolev type spaces of
vector-valued functions, defined via their symmetric gradient, is proposed. The
Sobolev spaces in question are built upon general rearrangement-invariant
norms. Optimal target spaces in the relevant embeddings are determined within
the class of all rearrangement-invariant spaces. In particular, all symmetric
gradient Sobolev embeddings into rearrangementinvariant target spaces are shown
to be equivalent to the corresponding embeddings for the full gradient built
upon the same spaces. A sharp condition for embeddings into spaces of uniformly
continuous functions, and their optimal targets, are also exhibited. By
contrast, these embeddings may be weaker than the corresponding ones for the
full gradient. Related results, of independent interest in the theory symmetric
gradient Sobolev spaces, are established. They include global approximation and
extension theorems under minimal assumptions on the domain. A formula for the
K-functional, which is pivotal for our method based on reduction to
one-dimensional inequalities, is provided as well. The case of symmetric
gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum
mechanics driven by nonlinearities of non-power type, is especially focused.
