  Cylindrical interfaces occur in sheared or deformed emulsions and as
biological or technological lipid monolayer or bilayer tubules. Like the
corresponding spherical droplets and vesicles, these cylinder-like surfaces may
host orientaional order with $n$-fold rotational symmetry, for example in the
positions of lipid molecules or of spherical nanoparticles. We examine how that
order interacts with and induces shape modulations of cylindrical interfaces.
While on spherical droplets $2n$ topological defects necessarily exist and can
induce icosahedral droplet shapes, the cylindrical topology is compatible with
a defect-free patterning. Nevertheless, once a modulation is introduced by a
mechanism such as spontaneous curvature, nontrivial patterns of order,
including ones with excess defects, emerge and have nonlinear effects on the
shape of the tube. Examining the equilibrium energetics of the system
analytically and with a lattice-based Markov chain Monte Carlo simulation, we
predict low-temperature morphologies of modulated cylindrical interfaces
hosting orientational order. A shape modulation induces a banded pattern of
alternatingly isotropic and ordered interfacial material. Furthermore
cylindrical systems can be divided into Type I, without defects, and Type II,
which go through a spectrum of defect states with up to $4n$ excess defects.
The character of the curvature-induced shape transition from unmodulated to
modulated cylinders is continuous or discontinuous accordingly.
