We study the gluonic color-spin locked (GCSL) phase in dense two-flavor quark matter. In this phase, the color and spatial rotational symmetries are spontaneously broken down to SO(2)_{diag} with the generator being an appropriate linear combination of the color and rotational ones. The Meissner masses of gluons and the mass of the radial mode of the diquark field in the GCSL phase are calculated and it is shown that this phase is free from the chromomagnetic and Sarma instabilities in the whole parameter region where it exists. The GCSL phase describes an anisotropic color and electromagnetic superconducting medium. Because most of the initial symmetries in this phase are spontaneously broken, its dynamics is very rich.

5 pages, 4 figures, clarifications added, to appear in Phys. Rev. D (Rapid Communications)