What does orthogonality of connection mean in coordinate way?
As I understand, a connection $\nabla: \Lambda^1M \rightarrow \Lambda^1M \otimes \Lambda^1M$ is torsion-free iff in any local coordinates $x_i$ with $[\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}]=0$ its Christoffel symbols has symmetry $\Gamma^l{}_{jk} = \Gamma^l{}_{kj}$. So I suppose, the semisum of any orthogonal connection with its "reversion" will be Levi-Civita connection.
But what is necessary for connection to be orthogonal, i.e. $\nabla(g)=0$ for $g$ - Riemann metric on $M$?
This means that the metric is compatible with the connection in the following sense: $$X g(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$$ (which follows easily from $\nabla g = 0$).