η + a0 + η + a1 + η + a2 + η . . . where each ai consists of ai elements linearly ordered then such a set has a strong η-representation. When A can be represented by such a computable linear order but the blocks ai may occur in any order and repetitions of blocks of the same size allowed, then A has an η-representation. And when A can be represented by such a computable linear ordering where the blocks ai may occur in any order, but only one block of size ai may occur, then A has a unique ηrepresentation. A Turing degree has a (strong, unique) η-representation if some set in that degree has one. This paper answers several questions on these representations which have been open since the early days in the investigation of computable linear orders.