Right arm spinners are either off spinners or leg spinners. But left arm spinners are refered to as chinaman and orthodox. They are never refered to as off spinners or leg spinners. Why are left arm spinners called differently?
Why are left arm spinners called differently?
To appropriately describe how the ball is bowled (right-handed or left-handed).
For right-arm bowlers, off spinners spin from off to leg and leg spinners spin from leg to off in right-handed batsmen.
For left-arm bowlers, "off spinners" spin from off to leg and "leg spinners" spin from leg to off only for left-handed batsmen, which is opposite of what is stated above. The left-arm orthodox is a type of finger leg-spin but is referred to as an off spin to a left-handed batsman as it turns the ball into him while the left-arm chinaman is a type of wrist off spin but is referred to as a leg spin as it turns away from a left-handed batsman.
Left-arm off spinners are known as "left-arm orthodox." It spins from off to leg to a left-handed batsman, is bowled left-armed, and uses a finger leg spin spin technique. (1)
A leg spinner also spins from leg to off, but is bowled right-armed and uses a wrist leg spin technique. (1)
As you can see, calling a spinner by how it spins only describes such. It does not describe how it was bowled (the bowler's handedness).
It is ultimately a naming convention for convenience.
There are by nature more right-handers, both bowlers and batsmen, so it makes some sense that the default, most compact terminology -- offspinner and legspinner -- is made relative to that combination.
Also, there are more finger spinners than wrist spinners (by nature?), so it makes some sense that among left armers the default, most compact terminology is assigned to the finger spinners.
Also, arguably, bowling left-arm wrist spin is less likely to be successful as a career, because it's harder to do than finger spinning, and you're effectively bowling offspin to the mostly right-handed batsmen, which is less difficult to deal with than legspin. So as a result of this being on average the least natural way to bowl and possibly the least effective, left-arm wrist spinners are so rare at the top level that they get the most obscure term.
If you want to use a methodical and neutral terminology, you can refer to any traditional spin bowler as right/left-arm finger/wrist spin, but try saying that all day long. And nowadays, with the rise of "mystery spinners", this categorization is starting to break down, and you will see bowlers listed as "leg break/googly" or some such.
(A similar asymmetry exists in baseball, where left-handers are referred to as "southpaws", but there is no common special term for right-handers.)