We present new spectroscopic identifications of 12 X-shaped radio galaxies and use the spectral data to derive starburst histories and masses of the nuclear supermassive black holes in these galaxies. The observations were done with the 2.1-m telescope of the Observatorio Astronómico Nacional at San Pedro Mártir, México. The new spectroscopic results extend the sample of X-shaped radio galaxies studied with optical spectroscopy. We show that the combined sample of the X-shaped radio galaxies has statistically higher black-hole masses and older episodes of star formation than a control sample of canonical double-lobed radio sources with similar redshifts and luminosities. The data reveal enhanced star-formation activity in the X-shaped sample on the timescales expected in galactic mergers. We discuss the results obtained in the framework of the merger scenario.