The questions have been raised: Why 13? Why 20? And what does the 13 x 20 = 260 equation mean?
I don't believe that any of these questions have a single answer. Here are some of the numerous answers I have acquired over the last 20 years.
20: This is the easiest one to understand. As Bruce points out, 20 is the number of digits on our hands and feet. The Mayan word for a count of 20 is "uinal," while the word for "human being" is "uinic." Obviously, the two are related. Also, Bruce is correct when he remarks that it might have something to do with the Moon. There is an oral tradition among the Maya that the 20th day of the moon is particularly auspicious, though I am not sure why. The hieroglyphic symbol for "moon" is the same as the one for the number "20". Only context can determine which meaning is intended. One of the problems with the Mesoamerican Calendar is that any question as to what this or that factor "means" has to be predicated on the question: "Among which culture and at what period of history?" There seems to have been a lot of variation. Shortly before the Spanish Conquest, the Yucatec Maya almost certainly believed that the Moon Goddess was the divine patroness of the Sacred Calendar.
13: The answer to this one is guaranteed to drive you nuts! First, we are back in the realm of the Moon Goddess. Martin Prechtel, who was the source for most of the anthropology emerging from the Lake Atitlan region during the 1970s and 80s, has remarked that the Tzutujil Maya recognize 12 aspects of the Moon Goddess with her totality comprising the 13th. Also, it should be noted that there are 13 major joints in the human body -- ankles, knees, hips, wrists, elbows, shoulders, and the neck. Also, while many Mesoamerican societies perceived the "creature at the base of the World Tree" as a caiman or alligator, they were also familiar with it as a turtle. If you ignore the weird little bumps and protrusions and just count the rectangles on a turtle's shell, you will find that there are 13 of them. Statuettes of turtles depicted as calendars go back all the way to the pre-Classic era in Mesoamerica. So 13 is also the number of Turtle Island. The Mayan "place of creation" was in the Gemini/Orion region, perceived as a turtle.
13 x 20 = 260. Some have remarked that 260 is the interval between solar zenith passages at 15 degrees north latitude. They have suggested that this places the original location of the Calendar's invention. One of the places suggested for its origin is Izapa, which is indeed at 15 north. But there is also good evidence that the Calendar was an Olmec invention, in which case we can forget about the zenith passage, since this is not at 15 north. Martin told me in private conversation that he didn't think that had anything to do with it. I asked him also about the factor mentioned by Bruce, that this is sometimes linked with the human gestation cycle. He replied, "That's a very basic level, like a gingerbread tale. We give that answer to anthropologists so that they will leave us alone and let us get back to work. There is always more than one answer to these questions, and the real story is much too long. It has a lot to do with Venus and the Moon."
Those remarks are from my notes, dated 1995. I never knew what he meant by Venus and the Moon. In the past year or two, I have discovered (through Susan Milbrath) that if you note the first conjunction that Venus makes with the Moon after its emergence from the Underworld, then pass through the visibility cycle, the next disappearance, and count forward to its next lunar conjunction after re-emergence, the (variable) interval will average out at around 260 days.