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Grid and e-science infrastructure interoperability is an increasing demand for Grid applications but interoperability based on common open standards adopted by Grid middle-wares are only starting to emerge on Grid infrastructures and are not broadly provided today. In earlier work we have shown how open standards can be improved by lessons learned from cross-Grid applications that require access to both, High Throughput Computing (HTC) resources as well as High Performance Computing (HPC) resources. This paper provides more insights in several concepts with a particular focus on effectively describing Grid job descriptions in order to satisfy the demands of e-scientists and their cross-Grid applications. Based on lessons learned over years gained with interoperability setups between production Grids such as EGEE, DEISA, and NorduGrid, we illustrate how common open Grid standards (i.e. JSDL and GLUE2) can take cross-Grid application experience into account.