We construct conforming finite elements for the spaces H(symCurl) and H(devsymCurl). Those are spaces of matrix-valued functions with symmetric or deviatoric-symmetric Curl in a Lebesgue space, and they appear in various models of nonstandard solid mechanics. The finite elements are not H(Curl)-conforming. We show the construction, prove conformity and unisolvence, and point out optimal approximation error bounds.