We present a new construction of a class pseudo BL-algebras, called kite
pseudo BL-algebras. We start with a basic pseudo hoop A. Using two injective
mappings from one set, J, into the second one, I, and with an identical
copy A with the reverse order we construct a pseudo BL-algebra
where the lower part is of the form (A)J and the upper one is
AI. Starting with a basic commutative hoop we can obtain even a
non-commutative pseudo BL-algebra or a pseudo MV-algebra, or an algebra with
non-commuting negations. We describe the construction, subdirect irreducible
kite pseudo BL-algebras and their classification