We study a model of dilute oriented loops on the square lattice, where each loop is compatible with a fixed, alternating orientation of the lattice edges. This implies that loop strands are not allowed to go straight at vertices, and results in an enhancement of the usual ${\rm{O}}(n)$ symmetry to ${\rm{U}}(n)$. The corresponding transfer matrix acts on a number of representations (standard modules) that grows exponentially with the system size. We derive their dimension and those of the centralizer by both combinatorial and algebraic techniques. A mapping onto a field theory permits us to identify the conformal field theory governing the critical range, $n\leqslant 1$. We establish the phase diagram and the critical exponents of low-energy excitations. For generic n, there is a critical line in the universality class of the dilute ${\rm{O}}(2n)$ model, terminating in an ${\rm{SU}}(n+1)$ point. The case n = 1 maps onto the critical line of the six-vertex model, along which exponents vary continuously.