how can I pin point my location on the opposite side of the world without sticking a needle through a globe?

From Wikipedia:

“The antipodes of any place on the Earth is the place which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the Earth and forms a true diameter. For example, the antipodes of New Zealand’s north island lies in Spain. Most of the earth’s land surfaces have ocean at its antipodes, this being a consequence of most land being in the northern hemisphere.”

Find your antipode here

One could just use latitude and longitude to find the spot on the opposite side. I.e. Longitude +180 degrees, latitude x -1 (same value but opposite side of equator). So if I am at Longitude 33 degrees, latitude 51 degrees north, the antipode would be at longitude (33+180=) 213 degrees, latitude 51 degrees south.