  In the present paper, we associate the techniques of the Lewy-Pinchuk
reflection principle with the Behnke-Sommer continuity principle. Extending a
so-called reflection function to a parameterized congruence of Segre varieties,
we are led to studying the envelope of holomorphy of a certain domain covered
by a smooth Levi-flat ``hat''. In our main theorem, we show that every
C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real
analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M'
is holomorphically nondegenerate. More generally, we establish that the
reflection function R_h' associated to such a C^infty-smooth CR diffeomorphism
between two globally minimal hypersurfaces in C^n (n > 1) always extends
holomorphically to a neighborhood of the graph of \bar{h} in M\times \bar{M}',
without any nondegeneracy condition on M'. This gives a new version of the
Schwarz symmetry principle in several complex variables. Finally, as an
appendix, we show that every C^infty-smooth CR mapping h: M to M' between two
real analytic hypersurfaces containing no complex curves is real analytic at
every point of M, without any rank condition on h. This answers a conjecture,
explicitely stated for C^infty-smooth maps at p.~328 of the survey article:
S.M. Baouendi, P. Ebenfelt and L.-P. Rothschild, Bull. Amer. Math. Soc. (N.S.)
37 (2000), no. 3, 309--336. Importantly, notice that a much stronger result,
valid for continuous maps, has been obtained by K. Diederich and S. Pinchuk in:
Michigan Math. J. {\bf 51} (2003), no. 1, 111--140; no.~3, 667--668.
