Definitions. A (Schauder) basis for a Banach space is called symmetric if it is unconditional and uniformly equivalent to all its permutations. It is called subsymmetric if it is unconditional and uniformly equivalent to all its subsequences. A generous introduction to symmetric and subsymmetric bases can be found in chapter 3 of the Lindenstrauss-Tzafriri book Classical Banach Spaces I.
Question 1. Does there exist a Banach space admitting a basis which is symmetric and another basis which is subsymmetric but not symmetric?
Question 2. Do we know an example of a Banach space which admits a subsymmetric basis but not a symmetric one?
Question 3. Do we know an example of a Banach space which admits a unique subsymmetric basis (up to equivalence) but not a symmetric one?
Discussion. I have actually already answered questions #2 and 3 in the affirmative, but I am wondering if this is already known. I do not have an answer for #1.