When you see glacial scratching that is highly direction, it is usually on a piece of bedrock or a large immovable boulder that the glacier has passed over. If you see it on a cobble and it’s on one surface only then that’s normally a piece that has subsequently broken away from the bedrock. On a small stone that has been dragged along below the glacier as rubble, the scratching will frequently be multidirectional and on multiple surfaces. “Cris-crossing” of the scars is not at all unusual in such circumstances and these stones also frequently exhibit “pseudo-faceting” – especially around the edges:
Glacial Scratching & pseudo-faceted edges on cobble from W Pennsylvania [pic by Callan Bentley]
I think the guys are right. That does look like glacial scarring arising from rubble being carried along at the interface between the base of the glacier and the bedrock it passed over. It’s pretty deep, but sometimes the smaller the stone, the more scarred it gets and post-glacial weathering can increase the depth. As Ray says, there might be some additional plow-scarring too.