I hear people accusing themselves and others quite often of being either a sadist or a masochist. Sometimes the terms are used correctly (on the surface, not in principle). Sometimes they are turned around or used backwards.
Sadist: An individual who derives pleasure from inflicting pain upon others. Though sadism is usually related to sexual pleasure, there is a sub-set of sadism that revolves around joy in knowing that you have caused others to suffer. Sexual sadists often begin simply--spanking, nibbling, slapping--in order to "spice up" their sexual encounters. As they progress, they may find that they have difficulty becoming aroused or achieving orgasm unless they seen true pain in the eyes of their partners. If left unchecked, a sadist, sexual or otherwise, could become addicted to the pleasure they derive from inflicting pain. This may lead them to commit unspeakable crimes or acts of torture.
Masochist: An individual who derives pleasure from receiving pain from others. Again, masochism as a psychological term primarily refers to the paraphilia--abnormal and/or unsanctioned sexual behaviors. In this case, a sexual masochist may begin with role-playing similar to the sadist, only on the receiving end of the spankings, nibbling, etc. These behaviors become a psychological condition when the masochist finds that they have difficulty becoming aroused or achieving orgasm unless they experience some form of physical pain. The severe masochist may experience irreparable harm at the hands of their partners, especially if they become involved with an unchecked sadist. The results could be devastating for the masochist and overly encouraging for the sadist. A non-sexual masochist may find that they seek pain to help them "feel alive." Masochism should not be confused with cutting or other forms of self-mutilation, as masochism involves activation of the pleasure center of the brain but cutting does not.
Cutting (self-mutilation): The act of causing physical harm, usually leaving visible scarring, to one's self. Unlike masochism, there is no sexual component to self-harm or self-mutilation. The individual does not typically experience the rush of positive endorphins that we associate with pleasure. Instead, much self-mutilation is an act of penance in which the individual feels that they have done something wrong or are not worthy of something, so they harm themselves in an attempt to atone for it. They may cut or burn themselves enough to leave a scar as a punishment and a visible reminder. Again, there is no pleasure associated with these acts, only pain.
Here are a couple things to keep in mind. A mean person is not necessarily a sadist. They may be lashing out to cover up some other feelings of insecurity. A "glutton for punishment"--someone who seems to have a hard time saying "no" or who tends to take on challenges that may go beyond their abilities--is not necessarily a masochist. They may have a need to be challenged or they may feel too insecure to turn down a request from someone. Bondage and domineering can be healthy sexual role-playing behaviors when adequate safeguards (safe words, mutual respect, time spent afterward reassuring the dominated partner of their worth) are in place. Self-harm is a serious cry for help. Although not a complete guarantee, many cutters end up attempting suicide at least once before anyone realizes that they need help. They tend to be really good at hiding the evidence of their self-mutilation so that others cannot detect the growing depression or anxiety that lead to the harmful behaviors.
As for the sadistic masochist, the two concepts cancel out each other.
Even with a split personality, receiving pain yourself is not causing physical pain to someone else, outside of the realms of science fiction.