  We study the two-dimensional Dirac operator with an arbitrary combination of
electrostatic and Lorentz scalar $\delta$-interactions of constant strengths
supported on a smooth closed curve. For any combination of the coupling
constants a rigorous description of the self-adjoint realizations of the
operators is given and the qualitative spectral properties are described. The
analysis covers also all so-called critical combinations of coupling constants,
for which there is a loss of regularity in the operator domain. In this case,
if the mass is non-zero, the resulting operator has an additional point in the
essential spectrum, and the position of this point inside the central gap can
be made arbitrary by a suitable choice of the coupling constants. The analysis
is based on a combination of the extension theory of symmetric operators with a
detailed study of boundary integral operators viewed as periodic
pseudodifferential operators.
