Inverting a weave is hiding it once it has been finished and tied off, so that even another channeler cannot tell it is there. Rand learned inverting from Asmodean, and used it on a number of occasions, most notably to put traps around Callandor and to hide the access ter'angreal for the Choedan Kal. The Supergirls also learned it from Moggy.

Reversing a weave, on the other hand, is hiding the weave as it is being woven, so your opponent cannot tell you are in the act of channeling, or even that you are holding the Power. It's first mentioned in WH; Cyndane, Graendal and Demandred all use reversed webs during the Battle of Shadar Logoth. Verin observes Graendal using it:

Using the full strength of her circle, [Verin] wove her shield, and watched aghast as it rebounded. The woman was already embracing saidar, though no light shone around her, and she was immensely strong!
[WH: 35, With the Choedan Kal, 646-647]

As far as we know, only the Forsaken know about reversing, so far.