I do like Christmas Cactus (Schlumbergera russelliana, and hybrids), though actually your plant and mine are both technically Crab Cactus (Schlumbergera truncata, and hybrids), also known as Thanksgiving Cactus. However, they are usually in full bloom at Christmas, so perhaps the common name should change (It sort-of already has), though that will cause even more confusion, or at least long unnecessary explanations, such as this one.
My Schlumbergera truncata hybrid is most likely either 'Thor Olga' or 'Exotic Dancer'. It has large, red and white flowers. With the green stems (not leaves), it is even color-coordinated for Christmas.
I also have purple-pink Easter Cactus (Schlumbergera x graeseri), but that is another story.