In his dictionary On the Meaning of Words (De verborum significatu), the grammarian Sextus Pompeius Festus explains the dies religiosi (‘days of pious scruple’) as follows:
The dies religiosi are those on which it is regarded as illicit (nefas) to do anything that is not necessary. Of this character are the thirty-six so-called dies atri, the dies Alliensis, and the days on which the mundus is open.
By ‘doing anything’, Festus means the beginning of meaningful private and public undertakings, such as (for instance) opening a court case or beginning business dealings, but also sacrifices, especially when they are connected with the beginning of an undertaking. However, when undertakings have already begun, they can be continued.
Aulus Gellius treats the same subject in more detail in Attic Nights 5.17. Here is the text of the J. C. Rolfe translation (slightly modified):
Why the first days after the Kalends, Nones and Ides are considered dies atri; and why many avoid also the fourth day before the Kalends, Nones or Ides, on the ground as religiosum.
Verrius Flaccus, in the fourth book of his work On the Meaning of Words, writes that the days immediately following the Kalends, Nones and Ides, which the common people ignorantly call dies nefasti, are properly called, and considered, dies atri, for this reason:— “When the city,” he says, “had been recovered from the Senonian Gauls, Lucius Atilius stated in the senate that Quintus Sulpicius, tribune of the soldiers, when on the eve of fighting against the Gauls at the Allia, offered sacrifice in anticipation of that battle on the day after the Ides; that the army of the Roman people was thereupon cut to pieces, and three days later the whole city, except the Capitol, was taken. Also many other senators said that they remembered that whenever with a view to waging war a magistrate of the Roman people had sacrificed on the day after the Kalends, Nones or Ides, in the very next battle of that war the State had suffered disaster. Then the senate referred the matter to the pontiffs, that they might take what action they saw fit. The pontiffs decreed that no offering would properly be made on those days.”
Many also avoid the fourth day before the Kalends, Nones and Ides, as ill-omened (inominalis). It is often inquired whether any religious reason for that observance is recorded. I myself have found nothing in literature pertaining to that matter, except that Quintus Claudius Quadrigarius, in the fifth book of his Annals, says that the prodigious slaughter of the battle of Cannae occurred on the fourth day before the Nones of August.
In summary: the days after (dies postriduani) the Kalends, Nones and Ides (including the infamous dies Alliensis or ‘day of the battle of Allia’) are dies atri, ‘black days’ or ‘unlucky days’. These atri belong to a larger class of dies religiosi, which also includes the three days of the year on which the mundus was opened, and by some accounts, the fourth day before Kalends/Nones/Ides – or rather, in modern English terms, the third day, as the Romans counted inclusively. Now, the openings of the mundus (and some other occasions mentioned by other sources) are rituals that no longer takes place, so that we may ignore them here. Taking just the days that follow from the system of Kalends, Nones and Ides, we get the following:
December 29th (‘fourth’ day before the Kalends of January)
January 2nd (day after the Kalends of January)
January 2nd (‘fourth’ day before the Nones of January)
January 6th (day after the Nones of January)
January 10th (‘fourth’ day before the Ides of January)
January 14th (day after the Ides of January)
For February: January 29th, February 2nd, 2nd, 6th, 10th, 14th.
For March: February 27th (28th in a leap year), March 2nd, 4th, 8th, 12th, 16th.
For April: March 29th, April 2nd, 2nd, 6th, 10th, 14th.
For May: April 28th, May 2nd, 4th, 8th, 12th, 16th.
For June: May 29th , June 2nd, 2nd, 6th, 10th, 14th.
For July: June 28th, July 2nd, 4th, 8th, 12th, 16th.
For August: July 29th, August 2nd, 2nd, 6th, 10th, 14th.
For September: August 29th, September 2nd, 2nd, 6th, 10th, 14th.
For October: September 28th, October 2nd, 4th, 8th, 12th, 16th.
For November: October 29th, November 2nd, 2nd, 6th, 10th, 14th.
For December: November 28th, December 2nd, 2nd, 6th, 10th, 14th.