  We are concerned with random walks on $\mathbb{Z}^d$, $d\geq 3$, in an i.i.d.
random environment with transition probabilities $\epsilon$-close to those of
simple random walk. We assume that the environment is balanced in one fixed
coordinate direction, and invariant under reflection in the coordinate
hyperplanes. The invariance condition was used in Baur and Bolthausen (2014) as
a weaker replacement of isotropy to study exit distributions. We obtain precise
results on mean sojourn times in large balls and prove a quenched invariance
principle, showing that for almost all environments, the random walk converges
under diffusive rescaling to a Brownian motion with a deterministic (diagonal)
diffusion matrix. We also give a concrete description of the diffusion matrix.
Our work extends the results of Lawler (1982), where it is assumed that the
environment is balanced in all coordinate directions.
