The class of Linear Inversion Transduction Grammars (LITGs) is introduced, and used to induce a word alignment over a parallel corpus. We show that alignment via Stochastic Bracketing LITGs is considerably faster than Stochastic Bracketing ITGs, while still yielding alignments superior to the widelyused heuristic of intersecting bidirectional IBM alignments. Performance is measured as the translation quality of a phrase-based machine translation system built upon the word alignments, and an improvementof 2.85 BLEU points over baseline is noted for French