Step 1: The central difference operator is defined as $\delta f(x_0) = f(x_0 + h/2) - f(x_0 - h/2)$, where $h$ is the interval of differencing. This operator requires values on both sides of $x_0$ to exist in the table.
Step 2: This is different from the forward difference operator $\Delta f(x_0) = f(x_0+h) - f(x_0)$, which only needs values ahead of $x_0$ (so $x_0$ can be the first tabulated point), and the backward difference operator $\nabla f(x_0) = f(x_0) - f(x_0-h)$, which only needs values behind $x_0$ (so $x_0$ can be the last tabulated point).
Step 3: Because the central operator needs entries on both sides, $x_0$ must sit strictly inside the tabulated range, that is, at some intermediary point of the series, not at either endpoint.
Step 4: This confirms that a central difference table takes its origin at an intermediary value of the series, matching option (C).
\[\boxed{\text{The origin } x_0 \text{ is the intermediary value in the series}}\]