This research article describes an orientation-invariant map in barrel cortex. Specifically, it found a map of whisked space in L2/3 and some possible mechanisms to do so. The map is centered on the whisking midpoint, so it is only invariant to whisking angle. The map changes based on whisking amplitude, so if that means the map stretches to fit the total angle whisked, then the map might only be invariant to whisking phase.
The mechanism to produce this map of scanned space requires multiple whiskers. As I understand it, when a whisker contacts the pole, it suppresses responses to subsequent contact by other whiskers, so the biggest response is in the barrel column corresponding to the first whisker to contact the pole during the whisking cycle, which depends on the place in scanned space.
This probably isn’t the entire mechanism, since the map of scanned space is continuous and they found this mechanism by removing whiskers.