We propose data driven score rank tests for univariate symmetry around a known center based on non-smooth functions. A choice of non-smooth functions is motivated by very special properties of a certain function on [0; 1] determined by a distribution which is responsible for its asymmetry. We modify recently introduced data driven penalty selection rules and apply Schwarz-type penalty as well. We prove basic asymptotic results for the test statistics. In a simulation study we compare the empirical behavior of the new tests with the data driven tests based on the Legendre basis and with the so-called hybrid test. We show good power behavior of the new tests often overcoming their competitors.
