Ok, Without fully computing out the orbits (Weekend project!), and JUST working with the periodicity of the months and days between cycles...
here's a tentative numerological/cosmological/astronomical framework.
First up, the clearly and profoundly significant numbers, and why they are so.
(1) The number one is the primary ordinal. The fundamental unit from which all others spring. It is the mythic counter balance and origin to the indivisible number 13. (See 13). There is but the one earth, and its ending is also its beginning.
(2) This number represents mystery, the raw power of the esoteric and arcane. To fully explain why, we need to explain the rest of the cosmology, so more on that later. in a nutshell, it represents the unknown and unknowable, and equal but opposite quantities that are only distinct when viewed side by side. It has manifestations EVERYWHERE. (A left hand, and a right hand. The mirror symmetries of living creatures, and reflections in mirrors. Etc.) It is also associated with works and wonders on a cosmic scale. (there are 2 spheres in the sky, The moon and the sun, etc.)
(5) This number represents a count of days, or a cycle of measure for mortal and mundane creatures and tasks. (i'd expect a base 5 number system for ordinary tasks, like book keeping, or base 10, with special magical significance added, based on 5 and 2.) It represents physical labors, and processes. (See the section on gods and their devisions below.)
(13) This is the indivisible number of completion. A full turning of the world, and a manifestation of the divine world beyond the physical one, which reflects it.
OK then.
First up, we have a year laid out like this:
3 months with 26 days, then one month with 25. This is followed by 5 months with 26 days, and then a month with 25, followed by 3 months of 26 days. This creates a full year.
Significance, and magical thinking mode engaged:
This places the first month of 25 days on or near the spring equinox, and the second month of 25 days on or near the fall equinox. There are 2 such times of the year.
Each normal month is 5 counts of 5 days, plus one holy day dedicated to the patron god of that month. Each god has 2 axes of manifestation, and the first is celebrated in the daylight hours, and the second after nightfall. So, EG, on the month of Granite, the god of "Wealth" is represented, and on the holy day, all the pleasures of excess that wealth brings are celebrated-- but after dark, the sobering realities of poverty and the pains it brings are given religious devotions. These months all fall during times of the year when the day and the night period are of uneven lengths, as is fitting for the gods they patronize on those months. (Some align more with a positive daytime influence, and other align more with a negative night time influence, making the time spent in the devotions to each apropos to their roles in dwarven civilization and cosmology.) This is magically represented as a full count of counts of days, (5 weeks, 5 days each, representing a full cycle of toil) in observance of 1 god, and its twin axes of the celestial mystery. It is important to point out that these are the lesser gods more associated with mundane events and occurrences. (Things involving mortals, or the mechanical nature of the mortal world.)
Each month sees a cycle from full moon, to new moon, to full moon again. In this cosmology, the full and new moons are not what are considered magical. Instead, it is the 2 periods of half moon that are given significance. (This is a masterwork pigtail cloak made of master work pigtail cloth made by Urist McClothier, Masterfully dyed midnight blue with dimple dye by Urist McDyer. On the item is an image of crescent moons in pearl. The crescent moons are the symbol of the local dwarven civilization. Etc.) This is because these two cycles always occur each month, and are both equal, yet opposite, despite being identical. (Both are 1/2 moon. BUT-- One is the left side white, and the other the right side white.) The new and full moon phases are simply the extrema heralded by the 2 half moon phases, and mark the beginning and end of the month.
There are 2 months of 25 days, in which no holy day is observed. Instead, the entire month is held special, and takes place when the first period of day and night are of equal duration. Like the magical period of the half moon (which occurs twice a month), this time occurs twice a year, and both have equal periods of day and night-- but one heralds the ascent of warmth and life, and the other of cold and desolation, and conversely, the counter to their opposite, while being equal quantities. These months hold much magical significance, and are dedicated to the opposite halves of the same deep mystery. The number of days in these months is a perfect count of a count of days (5 weeks of 5 days, even) and taken together, add up to 2 counts of counts of days (50 days), dedicated to the 2 equal, harbinger gods who share a single axis of mystery. (These gods are unknown, and unnamed. It is presumed one is male, and the other female, but which is which is not known, as is appropriate for their sphere of mystery. Only standing side by side can they be discerned one from the other.)
This gives 11 mundane gods each with 2 axes of expression, and 2 special gods of mystery that occupy an obscured single axis of power.
5 of those gods are strongly associated with beneficial mortal wellfare, which have months of devotion which align strongly with a daytime bias. (The summer months between the equinox months.) 5 of those gods strongly align with deleterious aspects of mortality, and have a strong nighttime bias. (Winter months.) At the winter solstice month, The most nighttime aligned god of mortal affairs is given devotions, The god of destruction and of fiery and bloody rebirth. (Armok, if you feel so inclined.
) This gives 2 counts of 5 gods, 2 gods of mystery, and 1 god of endings and origins, signaling a full turning of the cycle. (The god of destruction is also the god of creation; For through the destruction of the old is the new created, and the purpose of the new creation is to satisfy the will to destroy.) This makes 5 mundane gods of works of creation and benevolence, and 5 gods of decay and malevolence, 2 harbinger gods of equal mystery, and one god of origins and endings. A count of celestial days of creation, and a count of celestial days of destruction, driven by 2 hands of 1 will, making an indivisible circle of 13 points.
Thus, are there 13 gods for one planet, and through the expression of one for the other, both are fully manifested in their fullest entirety.
It needs more thought, and more refinements, but it very neatly deals with all of the "problems" of the 13 month model, and paints a very vivid mythic backdrop for a very unique culture, that fits VERY well with many already existing world features. (Is it merely coincidence that there are TWO truely fantastical metals in the world, and that both are buried deep, and hold deep mysteries? How about how one is the harbinger of great fortunes, and the other of certain doom?)
Isnt magical thinking fun?