How can a starting point south of the north pole to an endpoint north of the south pole be halfway around the world?
I think you are too into the numbers.   When talking about longitude and latitude they are two different things.  Longitude goes east and west from a point on the earth from 0 to 180 degrees east and 0 to 180 degrees west and describe semicircles of equal length running from the north pole to the south pole.  Latitude however is in degrees north or south of the equator and describe circular lines around the earth that get smaller as you get closer to the poles.  See this image:

Slicing along the lines of longitude would give you wedge shapes like when you think of orange slices.  Slicing along lines of latitude would give you circular slices of varying sizes.
To get an 'opposite' longitude you spin the earth on it's axis 180 degrees.  The opposite of 1 degree east is 179 degrees west.  The opposite of 10 degrees east is 170 degrees west.  So in your example it will be from 121.895 degrees west (longitude -121.895) to 58.105 degrees east (longitude 58.105).
The opposite latitude is also 180 degrees away.  However there are only 180 total degrees of latitude because their shapes are different.  Longitude goes from 0 to 180 degrees east and 0 to 180 degrees west giving a total of 360 degrees.  Since latitude represents a circle, there are only 180 degrees, from 0 to 90 degrees north and from 0 to 90 degrees south.  But you need to go a full 180 degrees.  The way it works is that it is just the same degrees,  just flip north and south.  Think about two points on opposite sides of the planet on the equator.  They are both at 0 degrees latitude.  If you move north 1 degree, the opposite will now be one degree south.  If you continue to 90 degrees north, the opposite is now 90 degrees south, which makes sense since the north and south pole are opposite.
So let's take Monterey, CA at 36.6 (positive is north), -121.895 (negative is west).  You cannot get to the opposite side of the earth by walking along a line of latitude because all lines of latitude except for the equator are not the right size.  If you divided the earth at that line of latitude you would not get two equal halves.  You can however get to the opposite side of the earth by walking along lines of longitude.  So say you start walking north.  After 53.4 degrees of latitude you arrive at the north pole.  Keep going in a straight line and you will start walking south on longitude 58.105 (positive is east).  After you reach the equator you will have travelled 143.4 degrees.  You have 36.6 degrees of walking south to go before you end up at -36.6 degrees latitude (negative is south).  You've travelled 180 degrees in a straight line to get to the opposite side of the earth and now you're in the Indian Ocean.