Interest in topological states of quantum matter has focused on the classification of topological but gapless electronic states, including Weyl and Dirac semimetals and various types of line node semimetals. Motivated by recently reported work on two-dimensional Cu$_2$Si which is a candidate 2D line-node semimetal, we study a model that supports a novel family of topological states on the primitive triangular lattice. Nontrivial k-space geometry emerges from propagation within an orbitally degenerate manifold (here the Si 3p-states) on a primitive lattice. We find that this model describes the expected line node degeneracies protected by a z-mirror symmetry, but the L=1 orbital symmetry also requires the presence of both linear and quadratic Weyl degeneracies at high symmetry points. Importantly in these latter cases compensating point singularities required by the global symmetry of the band structure for the L=1 orbital multiplet on the Bravais lattice are generically offset in energy. We augment this model with possible T- breaking perturbations to identify the observable consequences of this unusual k-space texture for various dissipationless transverse responses including an anomalous Hall conductance and a related orbital Hall conductance. We further consider a route to introducing these T-breaking fields on demand using optical fields and estimate strength of these perturbations for accessible intensities.