Let \(f = \sum_{i=0}^n A^{(i)}y_i\), \(A^{(n)}\ne 0\), be a linear homogeneous difference polynomial (LHDP) with coefficients in the difference field \(k\) of characteristic 0. The author defines \(f\) to be reducible at \(q\) to order \(r\), \(r 2\) there exists a class of second order difference equations reducible at \(q\) but not at any \(t1\). Proposition 3.1 is a more special result providing related information. From these results the author obtains a classification of reducible second order equations (proposition 3.3) and of third order equations reducible to order 2 (proposition 3.4). An example is given of a third order LHDP \(f\) reducible at \(q=3\) to order 2 but having no solution except 0 which satisfies a linear equation of order 2 over the coefficient field of \(f\).