  Completing the results obtained in a previous paper, we prove the symmetry of
Hadamard/Seeley-deWitt off-diagonal coefficients in smooth $D$-dimensional
Lorentzian manifolds. To this end, it is shown that, in any Lorentzian
manifold, a sort of ``local Wick rotation'' of the metric can be performed
provided the metric is a locally analytic function of the coordinates and the
coordinates are ``physical''. No time-like Killing field is necessary. Such a
local Wick rotation analytically continues the Lorentzian metric in a
neighborhood of any point, or, more generally, in a neighborhood of a
space-like (Cauchy) hypersurface, into a Riemannian metric. The continuation
locally preserves geodesically convex neighborhoods. In order to make rigorous
the procedure, the concept of a complex pseudo-Riemannian (not Hermitian or
K\"ahlerian) manifold is introduced and some features are analyzed. Using these
tools, the symmetry of Hadamard/Seeley-deWitt off-diagonal coefficients is
proven in Lorentzian analytical manifolds by analytical continuation of the
(symmetric) Riemannian heat-kernel coefficients. This continuation is performed
in geodesically convex neighborhoods in common with both the metrics. Then, the
symmetry is generalized to $C^\infty$ non analytic Lorentzian manifolds by
approximating Lorentzian $C^{\infty}$ metrics by analytic metrics in common
geodesically convex neighborhoods. The symmetry requirement plays a central
r\^{o}le in the point-splitting renormalization procedure of the one-loop
stress-energy tensor in curved spacetimes for Hadamard quantum states.
