Is there any Sharia basis for turning over an upturned shoe with the intention of earning reward from Allah, as is said, "Turn over the shoe for the sake of Allah, Most Glorious is He"?

There is no legal basis in Sharia for changing the orientation of an inverted shoe. Ibn Aqil considered it an act of the ignorant. Sheikh Ibn Uthaymin mentioned that the statement claiming angels do not enter a house with an inverted shoe is unfounded. As for changing the orientation of an inverted shoe, there is no harm in it, out of politeness and good taste, to avoid dirt, as the bottom of a shoe is often dirty.