You will be surprised to know, the calendar Inca invented and used were close to our current calendar. They used an approximately 365 days solar calendar or days, though the months started in December. Then they had also a lunar calendar, which was a 328 days year.
Two Calendars
As I just mentioned Inca had two calendars. One was a solar calendar or day time calendar, and the other one was lunar calendar or night time calendar.
The day time or Solar Calendar
The day time calendar was based on the solar cycle. They counted approximately 365 days in this calendar. This calendar was used for economic activities such as agriculture, mining, warfare and construction. This calendar was very important to Inca people, since they depend on this to fix the days on planting. They had towers in east and west of Cusco. They had some detail processes, where looking at the towers from certain location of main plaza of Cusco, they checked which day sun rose over the eastern tower and set on the western tower, to mark the day of planting corn and potatoes, and which day sun rose between the towers to mark the day of general sowing.
The night time or Lunar Calendar
This is the one used to mark the days of the all festivals. Since this calendar was based on moon it had only 328 days. Which meant twelve months of 27,33 days. Since lunar month is actually 28 days long, so it matches with the lunar months.
So they had around 37 days of difference between the two calendars. It is still not known, whether this difference was of any importance to the Incas.
Inca months
Now lets discuss the months in Inca calendar. Here I will discuss both the day time and night time calendar.
First Month
Present day calendar : December
Inca Lunar Month : Capaq Raymi
Inca solar month : Coca Planting
Their first month was actually started in our present day December. It was called Capaq Raymi, as the Capaq Raymi festival took place in this month. Capaq Raymi means Great Feast of Sun. They sacrificed gold, silver and lives to Sun. Capaq Raymi was marked with their Lunar calendar. As per solar Calendar this was the month of Coca Planting.
Few other rituals took place in this month. Boy's puberty ceremony was held in this month. Tribute from provinces was brought for Inca state and religion to Cusco at this month.
Second Month
Present day calendar : January
Inca Lunar Month : Camay Quilla
Inca solar month : Weeding of fields
Inca did their fasting and penance on this month. They offered sacrifices to Viracocha and covered their bodies with ashes. Also some parts of puberty ceremonies were carried to this month from the previous month.
Third Month
Present day calendar : February
Inca Lunar Month : Hatun-pucuy
Inca solar month : Harvest of potato and other root crops
Ceremonies were held to increase the corn and other crops.
Fourth Month
Present day calendar : March
Inca Lunar Month : Pacha-pucuy
Inca solar month : Harvest of potato and other root crops
Ceremonies were held to ensure that the crops ripened properly. Other ceremonies were held where people deprived themselves of salt and fruit.
Fifth Month
Present day calendar : April
Inca Lunar Month : Ayruhua
Inca solar month : Protecting corn field from deer, foxes.
Ceremonies honoring the Inca king were held.
Sixth Month
Present day calendar : May
Inca Lunar Month : Ayruhua
Inca solar month : Corn harvested
They enjoyed the celebration of corn harvested. Feast of Aymara was enjoyed with singing, drinking and dancing.
Seventh Month
Present day calendar : June
Inca Lunar Month : Inti Raymi
Inca solar month : Large potatoes were harvested and others planted
They enjoyed the Inti Raymi festival. Inti Raymi festival was the biggest festival of the year. People from all over Inca empire moved to Cusco during this festival. This festival is staged every year in present days.
Eighth Month
Present day calendar : July
Inca Lunar Month : Chahua - huarquiz
Inca solar month : Storage of potatoes and other crops
They enjoyed festival of harvest and ceremonies regarding irrigation.
Ninth Month
Present day calendar : August
Inca Lunar Month : Yapaquiz
Inca solar month : Planting of the corn and potato crops
Sacrificed were made to all the Gods in this month, specially to those who were related to force of nature. Feast of tilling was conducted. Land was tilled for the next growing season.
Tenth Month
Present day calendar : September
Inca Lunar Month : Coya Raymi
Inca solar month : Planting of the corn and potato crops
This time great feast for moon was celebrated. Also this is the month of purification. Cusco is purified. In the process of purification nobles sacrificed. Sacred idols of conquered were brought to Cusco to pay their homage to King. Eleventh Month
Present day calendar : October
Inca Lunar Month : Kantaray
Inca solar month : nothing specific
This was the driest season of the year. So they held ceremonies where they prey to their God for rainfall.
Twelfth Month
Present day calendar : November
Inca Lunar Month : Ayamarca
Inca solar month : Irrigation of the corn fields
The festival of dead were conducted in this month. They brought the dead bodies to main square of the city. But it was not the the dead bodies. They actually carried the mummies of dead kings only. Mummies were dresses in valuable cloths. Then they were offered foods. People sang and danced around those mummies in main square.