A boundary inside the sign, not at its edge
Sign boundaries get all the attention, but each sign also divides into three faces of ten degrees, and those internal lines behave the same way. A date whose degree range straddles one of them belongs to different faces in different years.
Ten June is such a date, though only just. In the great majority of years the noon Sun is still short of twenty degrees and the second face applies. In the years at the top of the range it slips over, and the third face applies instead.
The difference is that nobody argues about internal lines, so they pass unremarked. A page will happily assign a decan to a date without checking whether the date sits close enough to a face boundary for the assignment to depend on the year.
Which side is more likely
The range runs from roughly eighteen point seven to twenty point one degrees, so most of it lies below the line. The years that cross are the late ones in the leap cycle, where the drift has accumulated before February pulls it back again.
Because the margin is thin, the birth hour can matter too. The Sun covers about a degree a day, so a birth late in the evening sits a fraction of a degree further along than one at dawn. In a borderline year that fraction is the whole question.
None of this touches the sign. Gemini runs to the solstice, roughly ten days after this date, and a spread of a degree and a half has no prospect of reaching it. The only thing in question is which ten degree band inside Gemini applies.
What changes if you cross
The second face of Gemini goes to Mars in the Chaldean order and to Venus through Libra in the triplicity count. The third face goes to the Sun in the Chaldean order and to Saturn through Aquarius in the other. The readings are not neighbours.
None of that alters the sign, the element or the sign ruler. Mercury governs the whole of Gemini regardless, and a Sun at nineteen degrees and one at twenty one are equally Gemini. The face is a subdivision, and it is read as a tint rather than a replacement.
Settling it, and what remains after
The widget above takes your birth year, hour and place, computes the solar longitude for that exact moment, and reports the degree along with the distance to the nearest boundary. On this date the nearest boundary of interest is the internal one at twenty degrees.
Once that is settled, the larger unknowns remain. The Moon can be in either of two signs on 10 June and the ascendant in any of the twelve, both of which move a reading further than the choice between two faces ever will.
It is also worth keeping the weights in proportion. The face is the finest of the solar subdivisions and the one traditions treat most lightly, so a year that lands close to the line is a curiosity about the position rather than a problem to be solved.
Questions people ask about this
Is 10 June the second or third decan of Gemini?
It depends on the birth year. In most years the noon Sun is a little short of twenty degrees, which keeps it in the second face, and in a few years it has just crossed into the third.
Does that change my sun sign?
Not at all. Both faces are inside Gemini. The sign is settled on this date in every year sampled from 1930 to 2035.
Which rulers are involved?
Mars or Venus for the second face, depending on the scheme, and the Sun or Saturn for the third. The Chaldean order gives the first of each pair and the triplicity count the second.
Can my birth hour move me across?
In a borderline year, yes. The Sun advances about a degree a day, so a late evening birth sits noticeably further along than an early morning one.
What degree is a 10 June Sun?
Between roughly eighteen point seven and twenty point one degrees of Gemini at noon, depending on the year in the leap cycle.
How much does a decan actually matter?
Less than the sign for most practitioners, and far less than the Moon or the ascendant. It is a way of distinguishing one Gemini birthday from another rather than a separate verdict.