" Around here, we simply "get on" a seesaw, but "get up on" is not unusual. By the same token, we "get off" or possibly "get down from". We use the same terminology for horses, I think, and that is no accident.
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stephenlearnerintransitiveYou have a misunderstanding. Both "Get up on the bed" and "Get up" are intransitive because "on the bed" is not a direct object.
stephenlearnerwithout mentioning the "bed"?Get up here! / Get up there!