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We have fabricated strain-sensitive spin valves on flexible substrates by utilizing the large magnetostrictive FeGa alloy to promote the strain sensitivity and the composite free layer of FeGa/FeCo to avoid the drastic reduction of giant magnetoresistance (GMR) ratio. This kind of spin valve (SV-FeGa/FeCo) displays a MR ratio about 5.9%, which is comparable to that of the conventional spin valve (SV-FeCo) with a single FeCo free layer. Different from the previously reported works on magnetostrictive spin valves, the SV-FeGa/FeCo displays an asymmetric strain dependent GMR behavior. Upon increasing the lateral strain, the MR ratio for the ascending branch decreases more quickly than that for the descending branch, which is ascribed to the formation of a spiraling spin structure around the FeGa/FeCo interface under the combined influences of both magnetic field and mechanical strain. A strain sensitivity of GF = 7.2 was achieved at a magnetic bias field of -30 Oe in flexible SV-FeGa/FeCo, which is significantly larger than that of SV-FeCo.