We identify a subalgebra ˆH+n of the extended affine Hecke algebra ˆHn of type A. The subalgebra ˆH+n is a u-analogue of the monoid algebra of Sn⋉ℤ≥0n and inherits a canonical basis from that of ˆHn. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient Rn1 of ˆH+n that is a u-analogue of the ring of coinvariants ℂ[y1,…,yn]/(e1,…,en) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element ∗π∈ˆH+n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that Rn1 has cellular quotients Rλ that are u-analogues of the Garsia-Procesi modules Rλ with left cells labeled by (a PAT version of) the λ -catabolizable tableaux.