If $\alpha$ is an ordinal, then the space of all ordinals less than or equal to $\alpha$ is a compact Hausdorff space when endowed with the order topology. Let $C(\alpha )$ be the space of all continuous real-valued functions defined on the ordinal interval $[0,\,\alpha ]$. We characterize the symmetric sequence spaces which embed into $C(\alpha )$ for some countable ordinal $\alpha$. A hierarchy $\left( {{E}_{\alpha }} \right)$ of symmetric sequence spaces is constructed so that, for each countable ordinal $\alpha$, ${{E}_{\alpha }}$ embeds into $C\left( {{\omega }^{{{\omega }^{\alpha }}}} \right)$, but does not embed into $C\left( {{\omega }^{{{\omega }^{\beta }}}} \right)$ for any $\beta \,<\,\alpha$.