  We consider the most general vacuum cylindrical spacetimes, which are defined
by two global, spacelike, commuting, non-hypersurface-orthogonal Killing vector
fields. The cylindrical waves in such spacetimes contain both + and $\times$
polarizations, and are thus said to be unpolarized. We show that there are no
trapped cylinders in the spacetime, and present a formal derivation of Thorne's
C-energy, based on a Hamiltonian reduction approach. Using the Brown-York
quasilocal energy prescription, we compute the actual physical energy (per unit
Killing length) of the system, which corresponds to the value of the
Hamiltonian that generates unit proper-time translations orthogonal to a given
fixed spatial boundary. The C-energy turns out to be a monotonic non-polynomial
function of the Brown-York quasilocal energy. Finally, we show that the
Brown-York energy at spatial infinity is related to an asymptotic deficit angle
in exactly the same manner as the specific mass of a straight cosmic string is
to the former.
