Yes, there is a collection of $9g - 9$ curves that suffices. See Expose Six and Appendix C of "Thurston’s Work on Surfaces" edited by Fathi, Laudenbach, and Poenaru.

Here is a very short sketch - the details are complicated. Suppose that $\Sigma$ is the given surface of genus $g$. Let $P = \{\gamma_i\}$ be a pants decomposition of $\Sigma$. We first recall the *Fenchel–Nielsen coordinates*. Suppose that $\sigma$ is a hyperbolic metric on $\Sigma$. For every $\gamma_i \in P$ we then have

- a hyperbolic length $\rm{len}(\gamma_i, \sigma)$ and
- a twist parameter $\rm{tw}(\gamma_i, \sigma)$.

The former is a positive real number and the second is a signed real number. Care must be taken to understand the twist as a signed real number and not just as a circle valued number. We can recover the twist from length data as follows. For each $i$, pick *dual* curves $\alpha_i$ and $\beta_i$ so that the triple of curves $\{\alpha_i, \beta_i, \gamma_i\}$ meet each other non-trivially and minimally, and meet the rest of $P$ minimally. Then the lengths of these $9g - 9$ curves recover the twist parameters.

We now can discuss the original question. Again we have $\Sigma$ and $P$. We now recall the *Dehn-Thurston coordinates*. Suppose that $\delta$ is a simple closed curve. For every $\gamma_i \in P$ we then have

- an intersection number $\rm{i}(\gamma_i, \delta)$ and
- a twist parameter $\rm{tw}(\gamma_i, \delta)$.

The former is a non-negative integer and the second is an integer. Care must be taken to understand the twist. It can be defined using from intersection data with dual curves, chosen as above.