I'm having an embarrassingly hard time reconciling some basic calculations that I think are correct (but given my confusion, I won't make a warranty) and the discrepancy in pheneomenology of the geodesics on an ellipsoid (which are rather complex) versus on a sphere (which are just arcs of great circles).
Here's the basic calculations first:
Let $g$ be (the matrix of) a Riemannian metric on a sub(psuedo)manifold of $\mathbb{R}^n$ relative to coordinates $x$. Let $R \in GL_n$ and $x \mapsto Rx =: x'$ be the corresponding linear transformation. The matrix of the metric relative to the coordinates $x'$ is then $g' = R^{-T}gR^{-1}$, and (eschewing the Einstein convention in order to write matrices conventionally) the Christoffel symbols transform as $$(\Gamma')_{ij}^k = \sum_{\ell m p} \frac{\partial x_\ell}{\partial x'_i} \frac{\partial x_p}{\partial x'_j} \frac{\partial x'_k}{\partial x_m} \Gamma_{\ell p}^m.$$ Now $\frac{\partial x'_i}{\partial x_j} = R_{ij}$ and similarly $\frac{\partial x_i}{\partial x'_j} = (R^{-1})_{ij}$, so the preceding becomes $$(\Gamma')_{ij}^k = \sum_{\ell m p} (R^{-1})_{\ell i} (R^{-1})_{pj} R_{km} \Gamma_{\ell p}^m.$$ Now the geodesic equation is $$\ddot x'_k = -\sum_{ij} (\Gamma')_{ij}^k \dot x'_i \dot x'_j$$ which after unpacking the transformations becomes $$ \begin{align} \sum_q R_{kq} \ddot x_q & = -\sum_{ij} (R^{-1})_{\ell i} (R^{-1})_{pj} R_{km} \Gamma_{\ell p}^m \cdot \sum_r R_{ir} \dot x_r \cdot \sum_s R_{js} \dot x_s \\ & = -\sum_{\ell m p} R_{km} \Gamma_{\ell p}^m \cdot \sum_{ir} (R^{-1})_{\ell i} R_{ir} \dot x_r \cdot \sum_{js} (R^{-1})_{pj} R_{js} \dot x_s \\ & = -\sum_{\ell m p} R_{km} \Gamma_{\ell p}^m \cdot \sum_r \delta_{\ell r} \dot x_r \cdot \sum_s \delta_{ps} \dot x_s \\ & = -\sum_{\ell m p} R_{km} \Gamma_{\ell p}^m \cdot \dot x_\ell \cdot \dot x_p. \end{align} $$ Multiplying by $R^{-1}$ on the left, we get $$ \begin{align} \ddot x_a & = -\sum_k (R^{-1})_{ak} \sum_{\ell m p} R_{km} \Gamma_{\ell p}^m \dot x_\ell \dot x_p \\ & = -\sum_{\ell m p} \sum_k (R^{-1})_{ak} R_{km} \Gamma_{\ell p}^m \dot x_\ell \dot x_p \\ & = -\sum_{\ell m p} \delta_{am} \Gamma_{\ell p}^m \dot x_\ell \dot x_p \\ & = -\sum_{\ell p} \Gamma_{\ell p}^a \dot x_\ell \dot x_p. \end{align} $$ This is just the geodesic equation in the original coordinates. Now of course the whole point of differential geometry is to study quantities that are ultimately independent of any choice of coordinates, so this doesn't seem weird.
BUT: an ellipsoid is just a linear (or affine, if one prefers) transformation of a sphere. And the geodesics on these two surfaces behave very differently.