We study asymptotic expansions of the smeared L2-traces Fexp(-t P^2) and
FPexp(-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
theta. Studying the theta-dependence of the above trace invariants,
theta-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.