The "perpendicular sections tied in for stability" are commonly called dead men, because they are buried like a dead man. I have searched and never been able to find a design procedure for the type of timber retaining wall you have described. I was interested, because they are very common in my area. Poor performance of this type of wall is also very common, especially for anything larger than a small garden wall. I have come to the conclusion that since there is no commonly known or accepted methodology for the structural design of this type of wall, then basically 100% of them were not designed by an engineer.
There is similar type of timber wall that can be designed using engineering principles and for which structural design and construction guidance is available if you search, and that is a timber crib wall. However, a timber crib wall requires at least twice as many timbers as the non-engineered dead man walls like you are dealing with, so they are not popular these days, and most residential wall builders don't even know what a crib wall is. A timber crib wall is basically a mass gravity wall that consists of a timber box (or crib) that gets filled with soil, or better stone.
The timber dead man walls would theoretically rely upon soil passive pressure and/or skin friction between the soil and the dead men to provide a tieback force to resist lateral soil pressure against the wall face, but I have never seen a detailed design for the load path by which forces would get into the dead men. The timbers at the wall face should be spiked together and the dead men should be spiked to the wall face. I have thought about trying to calc out the number and spacing of spikes that it would take to tie the wall face together and the number of spikes it would take to transfer forces into the dead men, but I have never attempted it. I have a feeling that connections to the dead men would be the limiting factor, and thus you would need a LOT of dead men, and this is why these types of walls don't perform well. They aren't adequately tied together at the face, and they don't have enough dead men anchors or enough connection capacity to the dead man anchors to resist the soil lateral forces on the face of the wall.