“par contre” doesn’t exist in written French, my teacher said.

Ah the subtle nuances of language! I believe that to understand how and when to use expressions or compound words such as these, one must delve into the basic meaning of the words that make up the expression.

Littré has a description of the main protagonists CEPENDANT, POURTANT, NÉANMOINS, TOUTEFOIS, and another of en revanche.

Since you are a native English speaker I’ll try to translate as much as possible. Please note, however, that the reasoning is my own; you note that dictionaries aren’t helping you very much!

• cependant: literally something like “while this [is/was happening], meanwhile”, originally with just that simple meaning, but later with the added meaning that concerns us here. Probably the same mechanism as “still”
• pourtant: literally “for so much [to the contrary]” – probably the same mechanism as “even so”
• toutefois: literally “every time” – maybe the same mechanism as “however”
• néanmoins: literally “no less” – seems to be the same mechanism as “nevertheless” and “nonetheless”
• en revanche: literally “in revenge” (though revanche is a much gentler “revenge” usually employed for games, such as a return match)
• par contre: literally “by against”, usually rendered in English by “in contrast”, “on the other hand”

The first four are employed interchangeably, and Littré calls it an example of synonymous meaning with different etymologies. I’m sure that most native speakers have never consciously analyzed the oh-so-minor variations in meaning that I have tried to render here. I have found a little forum post in which the use cases of some of the English words are discussed

The last two are also synonymous (well, those who say “par contre” should not be used say that “en revanche” should be used instead, while those who admit both say that the first should be used to express opposition — ).

As to the difference between the groups, I would say (really going out on an unsupported limb here) that the first is used to analyze an existing conflict or dilemma, while “en revanche” is used to express a back-and-forth list of arguments for-and-against (pros and cons). As for “par contre”, personally I use it to introduce an opposing point of view, and I’m not alone in that!