Quotients with unconditional bases Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotients having unconditional bases (was Argyros the first who proved that?). This strange phenomenon tempts me to ask 
whether every reflexive non-HI space has a quotient with an unconditional basis? 
My apologies if this is well-known.
 A: The space of Gowers and Maurey is reflexive and H.I. , and as proved by Ferenczi every of its quotients and its dual is also  H.I. (as  mentioned in the answers of Bill Johnson (giving the reference) and Kevin Beanland). The results of Argyros and Felouzis are answering to the question.
Moreover the remark  of Bill Johnson is to the point. S. Argyros and myself  provided a reflexive and unconditionally saturated Banach space $X$  with an H.I. dual $X^*$. Thus every quotient of   $X$ is indecomposable. Additionally we show that every quotient of $X$ has a  further quotient which is H.I.   
S. Argyros,  A. Tolias,  Indecomposability and unconditionality in duality. Geom. Funct. Anal. 14 (2004), no. 2, 247–282. 
A: Every quotient of the original Gowers-Maurey space is HI; Ferenczi proved this. Argyros-Felouzis produced examples of reflexive spaces that are not HI (e.g. contain some $\ell_p$) and have HI duals. Any of these spaces give an example of a space such that no quotient has an unconditional basis. I think it's safe to say that very little in the theory of HI spaces can be classified as 'well-known'.
A: The original Gowers-Maurey HI space GM is reflexive.  Ferenczi proved that the dual (and also  every quotient) of GM is HI. Then GM $\oplus$ GM is not HI but cannot have a quotient with unconditional basis, for then its dual would have a subspace with an unconditional basis.  But if $X\oplus X$ has a subspace with an unconditional basis, then also $X$ has a subspace with an unconditional basis (the two complementary projections onto the copies of $X$ in the direct sum $X \oplus X$ cannot both be strictly singular on the same subspace). 
I would guess that the experts even know that there is an unconditionally saturated reflexive space $X$ whose dual is HI and thus every quotient of $X$ cannot have an unconditional basis. 
Ferenczi, V. Quotient hereditarily indecomposable Banach spaces. Canad. J. Math. 51 (1999), no. 3, 566–584.
