Yes, in most situations, if there is no sign prohibiting the practice, you may turn right on red.  And, yes, there is such a thing as a left turn on red in situations where you are turning from a one way street on to another one way street.  The law is spelled out in Massachusetts General laws Chapter 89, Section 8 and says in relevant part:

At any intersection on ways, as defined in section one of chapter ninety, in which vehicular traffic is facing a steady red indication in a traffic control signal, the driver of a vehicle which is stopped as close as practicable at the entrance to the crosswalk or the near side of the intersections or, if none, then at the entrance to the intersection in obedience to such red or stop signal, may make either (1) a right turn or (2) if on a one-way street may make a left turn to another one-way street, but shall yield the right-of-way to pedestrians and other traffic proceeding as directed by the signal at said intersection, except that a city or town, subject to section two of chapter eighty-five, by rules, orders, ordinances, or by-laws, and the department of highways on state highways or on ways at their intersections with a state highway, may prohibit any such turns against a red or stop signal at any such intersection, and such prohibition shall be effective when a sign is erected at such intersection giving notice thereof. Any person who violates the provisions of this paragraph shall be punished by a fine of not less than thirty-five dollars.

Hope that is helpful.  For more information or to post a question, visit our Massachusetts Driving Law Discussion Forum.