  A rectangular graphyne sheet is composed of units similar to phenyl rings
that are linked by acetylenic chains, as in hexagonal $\gamma$-graphyne. This
system is organized over a rectangular lattice similar to that of the recently
synthesized biphenylene network. We investigate the stability of this sheet
from different perspectives and study its electronic structure. Rectangular
graphyne is a semiconducting system in its pristine form and features a pair of
highly localized states. These characteristics are correlated with the
structural anisotropy of the system, since its frontier states behave like
quasi--1D states embedded in the 2D lattice. We further consider modified
systems in which longer acetylenic links are introduced. We discuss how a
strategic choice of the position of these longer bridges can lead to specific
changes of the electronic structure of the rectangular graphyne sheet.
