This is the same question as asked here, but of course I think with a different twist.
A definition of orthogonal contrasts is given in another (great) answer to a popular question on Cross Validated by @ttnphns. Three conditions are needed: 1- balanced design 2- each row of the contrast matrix L sums to zero 3- rows are orthogonal.
According to this definition, Helmert contrasts are orthogonal, but effect contrasts are not. A partial answer to my question is given here: "traditional dummy coding is fine for regression coefficients, but since traditional dummy codes aren’t orthogonal, it messes things up when you’re just trying to partition variance (i.e. an ANOVA)". I understand some part of the variance can be explained by two or more non-orthogonal terms, so partitioning it might be in some sense arbitrary. However the explanation goes on demonstrating that using effect contrasts (non-orthogonal) produces the same ANOVA as using Helmert contrasts (orthogonal), while dummy coding (non-orthogonal) produces different, wrong results. I have tested this fact on other data and with other models as well. It seems the only needed condition for ANOVA is 2 above. Is that correct?
One perhaps related question: I generally have to do with GLMs (or mixed models) with >2 factors and find it handy (once I found that an interaction is significant) to write down post-hoc contrasts (i.e. customized contrasts/ linear combinations of the terms in the GLM) that might help interpret the meaning of the interaction in terms of coefficients in the dummy or in the effects coding. Is this ok, or should I always refer to an orthogonal coding scheme? What I suspect is that the CIs for arbitrary contrasts built as linear combinations of coefficients from the dummy (or perhaps even other non-orthogonal) coding schemes might be wrong.
