The complete list should be
and you can maybe add
\mid produces the same symbol as
\vert, but with the type “relation”; similarly
\colon produces a colon, but with a different spacing from
There is also
\slash, that produces
/ but with a line break allowed after it.
I'm not sure why
\ast has been defined in plain TeX (possibly because Knuth sometimes needed
* with a different meaning). The reason for
\lbrack is for providing
[ in a context where the latter cannot be used because it could be mistaken for the start of an optional argument;
\rbrack is there for symmetry. For sure, Knuth needed
| for his own purposes, so he provided
\vert as an alias.
amsmath the same symbol as
| is produced by
\rvert, but with the types “opening” and “closing” respectively.
The reason for
\Relbar is because the equals sign is used as extender for long double arrows; in some fonts a different symbol might be used, so
\Relbar is by default defined as
=, but can be redefined without the need to modify the definitions for
\Longleftarrow needn't be.
\rbrace are useful in order to fool smart editors that like to pair delimiters.
There is no alias for
-, but they can be easily defined.
A reversed situation happens for
\langle (and similarly for
\rangle): the constructions
\left< are shorthands for
< should never be used alone for denoting an angle bracket.