  We develop the elastic theory for inversion-asymmetric tethered membranes and
use it to identify and study their possible phases. Asymmetry in a tethered
membrane causes spontaneous curvature, which in general depends upon the local
in-plane dilation of the tethered network. This in turn leads to long-ranged
interactions between the local mean and Gaussian curvatures, which is not
present in symmetric tethered membranes. This interplay between asymmetry and
Gaussian curvature leads to a new {\em double-spiral} phase not found in
symmetric tethered membranes. At temperature $T=0$, tethered membranes of
arbitrarily large size are always rolled up tightly into a conjoined pair of
Archimedes' spirals. At finite $T$ this spiral structure swells up
significantly into algebraic spirals characterized by universal exponents which
we calculate. These spirals have long range orientational order, and are the
asymmetric analogs of statistically flat symmetric tethered membranes. We also
find that sufficiently strong asymmetry can trigger a structural instability
leading to crumpling of these membranes as well. This provides a new route to
crumpling for asymmetric tethered membranes. We calculate the maximum linear
extent $L_c$ beyond which the membrane crumples, and calculate the universal
dependence of $L_c$ on the membrane parameters. By tuning the asymmetry
parameter, $L_c$ can be continuously varied, implying a {\em scale-dependent}
crumpling. Our theory can be tested on controlled experiments on lipids with
artificial deposits of spectrin filaments, in-vitro experiments on
%\sout{artificial deposition of spectrin filaments on} red blood cell membrane
extracts, %\sout{after %depletion of adenosine-tri-phosphate molecules} and on
graphene coated on one side.
