Spacing commands for text mode can also be used in math mode, but not conversely, unless you load
amsmath (for a couple of them).
The LaTeX kernel defines
\mskip\thickmuskip respectively. For reference
\, is the space between an ordinary symbol and an operator or conversely (like in
2\sin x) or after a punctuation symbol;
\: is the space around a binary operation symbol (like in
\; is the space around a relation symbol (like in
However, the LaTeX kernel defines
\, to be also used in text mode, inserting a kern of width 0.1667em (3/18-th of an em). If
amsmath is loaded, also
\; can be used in text mode, inserting kerns of 0.2222em (4/18-th of an em) and 0.2777em (5/18-th of an em). These width are the same natural widths of the math mode spacings, expressed in
em units (but the em can differ).
Specific math mode spacing indeed uses
mu units, where 18mu is the same as 1em in the math symbol font. Such units can only be used as arguments to
\mkern, which is legal only in math mode. On the other hand, standard units and
\kern can appear in math mode as well as in text mode (and
em will refer to the font current when math mode was started).
This explains why
\qquad can be used in math mode: they will add 1em of space referring to the current text font. In some rare occasions
\hspace can turn out to be useful (notice the emphasis, though).
What commands are there for horizontal spacing?
Lengths and when to use them
Difference between \mskip and \mkern
How can we decide a glue or a kern when we are writing a macro?