For $n \gg 0$, we have that ${\rm dim~} R_n = {\rm dim~}  H^0(X, \mathcal O(n))$, where $X = {\rm Proj}(R) \subseteq \mathbb P(R_1^{\vee})$ is the projective scheme associated to $R$ and $\mathcal O(n)$ is the degree $n$ line bundle.      

In fact, we have that for all **integer** values that $$p_R(n) = \chi(H^*(X,\mathcal O(n))) := \sum_{i} (-1)^i {\rm dim} ~ H^i(X, \mathcal O(n)).$$

This gives an interpretation of $p_R(n)$ for negative values,  but it is somewhat abstract in general.  However in special cases (e.g. when $X$ is Cohen-Macaulay) there are more concrete interpretations of these Euler characteristics.

Let $K$ be a dualizing complex for $X$.  Then ${\rm Ext}^*_X(\mathcal O,\mathcal O(-n))$ is graded dual to $Ext^*(\mathcal O(-n),K) = H^*(X, K(n))$.  Thus in particular, if $X$ is Cohen-Macaulay of dimension $d$, so that $K = \omega_X[d]$  for some dualizing sheaf $\omega_X$, then we get that for all $n \gg 0$, that $$p_R(-n) = (-1)^d \chi(H^*(X, \omega_X(n)) = (-1)^d {\rm dim~} H^0(X, \omega_X(n)).$$ (We used that $n \gg 0$ in the last equality).  So in this case, the negative values of $p_R(n)$ can be interpreted as the graded dimensions of the dualizing module of $R$.  

As an example, you can take $R = k[x_0, \dots, x_d]$ so that $X = \mathbb P^d$ and $\omega_X = \mathcal O(-d-1)$ and one recovers the combinatorial reciprocity $${-n + d \choose d} =  (-1)^d {n -1 \choose d}.$$

Also this story is closely related to Ehrhart reciprocity, because Ehrhart functions are Hilbert functions on toric varieties.