I'm not clear on the question? Are you referring to opposing double left turn lanes operating under protected permissive phasing, or just opposing protected permissive phasing?
The Highway Capacity Manual (page 10-22)recommends protected left turn phasing if more than a single lane is being provided. I can not recall ever seeing a permissive double left turn lane at a traffic signal.
In Rhode Island we have used opposing protected permissive left turn phasing, but only if a left turn lane is provided.
In the absence of left turn lanes a protected move can only be provided in one direction.