Four of your months are numbers, and every one of them is two places off. Your calendar still counts from a New Year that was abolished two thousand years ago.
Documented
The numbers are the receipt
The oldest Roman year, tradition ascribes it to Romulus, ran ten months from March to December, roughly 304 days. Quintilis and Sextilis were simply Fifth and Sixth, and the last four months are still the Latin numbers seven through ten. Livy 1.19 and Plutarch's Life of Numa 18-19 record the reform: Rome's second king, Numa Pompilius, added January and February and stretched the year to 355 days. The count from March survived both the addition and everything after it.
Assessment: The numerical names are not a puzzle; they are an archaeological layer. You can read the original shape of the Roman year directly off your own wall calendar, the way you can read a coastline off a map of the sea floor.
Stories
The monthless winter
In the standard reconstruction, the ten-month year did not just start late; it stopped. After December came a stretch of uncounted winter, no months, no kalends, nothing to levy or plant, until the year woke up again in March. The calendar slept through the cold because a city of farmers and soldiers had no business to schedule in it.
Assessment: Whether the gap was a formal void or just unrecorded time is debated, but the structure is real: the calendar was an instrument for the working year, not a cage for all of time. The idea that every day must have a number is the later invention.
Interesting claims
January was an afterthought twice
Even after Numa added the winter months, the official year still opened in March. Only in 153 BC did the consuls begin taking office on January 1, and for purely administrative reasons: the Spanish war demanded the new commanders leave earlier. The civil year moved, the religious rhythms stayed, and the month-names stayed with them. Two thousand years later the seventh month is the ninth, in every language Rome ever touched.
Assessment: The renumbering of 153 BC is documented and decisive, which makes the survival of the old numbers stranger, not less. Empires repaint everything; this signpost was somehow beneath notice.
Sources and further reading
Livy, Ab Urbe Condita 1.19, Numa divides the year into twelve months
Plutarch, Life of Numa 18-19, the reform of Romulus' ten-month year
Varro, De Lingua Latina 6.33-34, the month names explained
Censorinus, De Die Natali 20-21, the Roman year-count
Macrobius, Saturnalia 1.12-16, the antiquarian dossier
Livy, Periochae 47, the consular year moves to January 1 in 153 BC
Michels, The Calendar of the Roman Republic (1967), the monthless winter reconstructed
Worth asking
If Rome renumbered the year twice and left the old numbers nailed to the doors, what else in your life is a measurement of a system that no longer exists?