We study asymptotic expansions of the smeared L ²-traces Fe^{−t P^2} and FPe^{−tP^2}, where P is an operator of Dirac type and F is an auxiliary smooth endomorphism. We impose chiral bag boundary conditions depending on an angle θ. Studying the θ-dependence of the above trace invariants, θ-independent pieces are identified. The associated stability theorems allow one to show the regularity of the eta function for the problem and to determine the most important heat kernel coefficient on a four dimensional manifold.