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.
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