The line at ten degrees
Each sign splits into three faces of ten degrees. Virgo's first runs from the start of the sign to ten, its second from ten to twenty. Across 1930 to 2035 the noon position on 2 September has fallen between about 9.1 and 10.5 degrees, which puts some years on one side of that line and some on the other.
So the honest statement is that this birthday belongs to the first decan of Virgo in some years and the second in others. It is not a blend and it is not undecided in principle. For any particular birth the Sun sits at one longitude and that longitude is either below ten degrees or above it.
What decides which
Two things. The year, because the leap cycle moves the Sun's position on a fixed date back and forth through a band a little over a degree wide, and the hour, because the Sun advances close to a degree a day and therefore a couple of arcminutes an hour. A birth early on 2 September in a western time zone and one late in the day in an eastern one are separated by enough motion to matter when the position is near the line.
The widget above resolves it. Give it the year, the time and the place and it returns the longitude to minutes of arc, which settles the face without guesswork.
Why the two faces read differently
The Chaldean order runs Mars, Sun, Venus, Mercury, Moon, Saturn and Jupiter in a loop from the start of Aries and never resets, so consecutive faces always have different rulers. Virgo's first falls to the Sun and its second to Venus. The triplicity method, walking the earth signs Taurus, Virgo and Capricorn in order, gives the first to Virgo under Mercury and the second to Capricorn under Saturn.
Tradition reads the solar face as the sign's work done in the open and the Venus face as the same care turned towards form and proportion, with the Saturn reading pulling instead towards structure and restraint. All of that is interpretation laid over a measured band, and none of it is better or worse to be born in.
It is also worth saying plainly that neither face is a better place to be born. Tradition attaches descriptions to them, not verdicts, and a reading that treats one ten degree band as fortunate has left the tradition behind.
The rest of the chart is untouched
None of this affects the sign. The Sun entered Virgo in late August and stays until the end of September, so 2 September is Virgo in every year, in every time zone, at every hour.
Nor does the face question compete with the real uncertainties. The Moon on this date is in one of two signs and only the hour says which. The Ascendant can be any of the twelve, changing roughly every two hours, and the houses follow it. The face is a fine detail on the one placement the date does fix, and the placements the date does not fix are the ones that make a chart personal.
Questions people ask about this
Is 2 September first or second decan of Virgo?
It depends on the year, and sometimes on the hour. Noon positions across the last century run from about 9.1 to 10.5 degrees, and the face boundary sits at ten, so both answers occur. The widget above settles yours.
Is the sign in doubt as well?
Not at all. Virgo is certain on this date in every year and every time zone. Only the ten degree face boundary inside the sign is genuinely year dependent.
What is the difference between the two faces?
The traditional ruler. Under the Chaldean order the first face answers to the Sun and the second to Venus. Under the triplicity order they answer to Mercury and Saturn respectively. The sign level reading is unchanged either way.
Do I need my birth time to know my decan?
If your year puts the noon position near ten degrees, yes, because the Sun moves a few arcminutes an hour and a birth at either end of the day can land on different sides.
Does the decan matter much in practice?
It is a secondary layer. Crossing it changes which planet traditionally colours the Sun's placement and nothing else. It does not change the sign, the element, the mode or any aspect.
What did the Sun do earlier that day?
It advanced a little under a degree over the twenty four hours, as it does through the whole of this part of the year. That is why a single date can span the boundary in some years.