He’s not left-wing but he’s not right wing either, not really. He’s a compromise guy. That’s one of his consistent features.
Right now, his compromise is between the standard Democrat and the Progressives, so he’s adding some policies that bring in Progressive issues that moderate Democrats and moderate Republicans (which ever ones are still there) won’t like.
But he’d just as easily straddle between moderate Democrat and slightly further right Republican if need be.
It’s just right now, the far right in the Republican party is in its own separate world right now, disconnected from the rest of things but always doing something.