As New York’s guide and curator of Indian dance’s ancient, thriving idioms, Rajika Puri has been indispensable. In her performance preludes, she enlightens and charms with summaries of the dance’s story and demonstrations of its defining gestures, so lay audiences can settle in and enjoy the work. Moreover, she resists the orthodoxy dominating Indian institutions in favor of master mavericks who stretch the mold. Puri’s final, 14th edition of “Dancing the Gods” (co-presented by the World Music Institute and Asia Society) features a different dancer-choreographer each night. All three programs should be extraordinary, but if you have to choose, one advantage of Indian-Americans Parul Shah and Mythili Prakash are the equally daring musicians they bring onstage. Shah foregoes kathak’s usual frontal approach for a voluptuous 360 degrees and a group consciousness that nods to “women who have been forgotten and women whose labor is often invisible.” Prakash pushes bharatanatyam’s reliable right angles until they splinter and flame. —Apollinaire Scherr