In psychology, there is a concept of Paradoxical Intervention. It is mentioned on the same the Wikipedia page you cited, but is perhaps more clearly defined by writer, William Hay:
In a paradoxical intervention the therapist tells a person to do something with the intention of achieving the exact opposite result. The most extreme example is the therapist telling a suicidal patient to kill themselves with the result that the patient becomes non suicidal.
This is clearly one form of "reverse psychology".
Does it work? This was examined in a meta-analysis of twelve studies in 1987:
Shoham-Salomon, Varda;Rosenthal, Robert Paradoxical interventions: A meta-analysis., Journal of Consulting and Clinical Psychology, Vol 55(1), Feb 1987, 22-28. doi: 10.1037/0022-006X.55.1.22
The effectiveness of paradoxical interventions in psychotherapy was evaluated in a meta-analysis of 12 data sets. Overall, paradoxical interventions were as effective as (but not more effective than) the typical treatment mode. However, paradoxical interventions showed relatively greater effectiveness than other interventions (a) 1 month after treatment termination and (b) with more severe cases.
So, yes, reverse psychology can work in a clinical setting.
(I am ignoring the more typical setting of dealing with recalcitrant children, because I think that has been shown to be somewhat effective in the short-term by virtually every parent.)