As visible on the Evans blog Refutation of Bruhn Dr. Evans does not grasp it. So I just posted a diagram at Connection & Torsion showing an asymmetric connection yielding a antisymmetric torsion. Hence the antisymmetry of torsion does NOT imply the antisymmetry of connection. And vanishing torsion is possible if and only if the connection is symmetric. The given example can readily be extended to higher dimensions.
Dr. Evans refers to the ''antisymmetry of the commutator''. The commutator can be found in S.M. Carroll's Lecture Notes on GR, p.75, Eq. (3.66).
[Ñμ,Ñν]Vρ = Rρσμν Vσ − TμνλÑλ Vρ Carroll: (3.66)
It evidently contains only the torsion coefficients Tμνλ which are double the antisymmetric part of the connection coefficients
Tμνλ := Γλμν − Γλνμ = 2Γλ[μν] . Carroll: (3.16)
The connection coefficients Γλμν are not tied by the commutator, only their antisymmetric part. Therefore no conclusion for the antisymmetry of the connection coefficients themselves is possible from the commutator Eq. (3.66). A non-antisymmetry, i.e. merely asymmetry, does not disturb the validity of Eq.(3.66) and is therefore compatible with the asymmetry of the commutator equation. See the example Connection & Torsion.