jmc

If $9$ is its own inverse $(\mathrm{mod}m)$, then $9\cdot 9\equiv 1(\mathrm{mod}m)$, or in other words, $m$ is a divisor of $9^2-1=80$.

But if $3$ is $\text{not}$ its own inverse $(\mathrm{mod}m)$, then $3\cdot 3\not\equiv 1(\mathrm{mod}m)$, so $m$ is not a divisor of $3^2-1=8$.

Thus, we wish to count divisors of $80$ which are not divisors of $8$. There are ten divisors of $80$: $$1,2,4,5,8,10,16,20,40,80.$$ Of these, six are not divisors of $8$: $$5,10,16,20,40,80.$$ Therefore, there are $\boxed{6}$ possible values of $m$.