Tides coming in is normal, but there are rivers that change direction for other reasons. The Chicago River and the Mississippi can change direction due to storm surges, for example (for the Chicago River, that temporarily returns it to its flow direction, into Lake Michigan (https://en.wikipedia.org/wiki/Chicago_River#Course))
“As the monsoon rain begins, the water level of the river begins to rise. As the water level continues to rise, the flow of the river reverses. The water level of the Tonlé Sap increases by about 10 metres (33 ft), the flow of the Mekong gradually decreases at the end of the rainy season, and the flow of the Tonlé Sap then reverses and began to replenish the flow of the Mekong.”
Even where it happens (it's common, the cases I know personally are Nantes or Antwerp) it's still obvious which side is right and which side is left, it's simply according to the main flow of the river.
Although in this instance, I think it's more common to talk about north/island/south in Nantes rather than left/right banks. While Antwerp's part on the left bank is literally called "left bank" (Linkeroever).
The Thames is tidal in London so the water does 'flow' both ways, but I'm fairly sure you'd say it flows towards the river but London is on a tidal section.
Isn't there some river where the flow goes either way depending on tide? I'm thinking it might be in Alaska, but there might be more than one.