Why 21 June was chosen as International Yoga Day?

Uttarayana & Dakshinayana:

The term Uttarayana is derived from two different Sanskrit words “uttara” (North) and “ayana” (movement) thus indicating a semantic of the northward movement of the Earth on the celestial sphere. This movement begins to occur a day after the winter solstice in December which occurs around 22 December and continues for a six-month period through to the summer solstice around June 21 (dates vary). This difference is because the solstices are continually precessing at a rate of 50 arcseconds / year due to the precession of the equinoxes, i.e. this difference is the difference between the sidereal and tropical zodiacs. The Surya Siddhanta bridges this difference by juxtaposing the four solstitial and equinotial points with four of the twelve boundaries of the rashis.

The complement of Uttarayana is Dakshinayana, i.e. the period between Karka sankranti and Makara Sankranti as per the sidereal zodiac and between the Summer solstice and Winter solstice as per the tropical zodiac.