Thanks for providing the extra detail, particularly the link to the article by Rich Apodaca. I now understand there is a difference between your definition of “round-trip” and the usual LO one (in terms of application). From the article:
The process of moving a chemical structure or drawing from a structure editor [external application] to a word processor [office suite] and back is called “round trip editing”.
This term is usually used in LO in relation to having an office document edited by application A (e.g., LO), then application B (e.g., MSO), then application A again. This is usually done using Microsoft file formats, but with MS Office 2013 can now be done more easily with ODF. The distinction is the editing of an entire compound file, rather than a single piece of that compound file.
There is a special section of the OOXML specification (17.17.3) dedicated to Roundtripping Alternate Content. Again though, I doubt this applies to an expectation of cut / paste roundtripping (although it may). There does not appear to be an equivalent section in the ODF specification v1.2 for handling content not defined by the spec. This may be the root cause of what you are experiencing.
 The idea of cutting and pasting an embedded object (e.g., drawing) back and forth between an office application and an external drawing application is (to me) a poor example of workflow. This is a workflow that Object Linking and Embedding (OLE) was specifically designed to solve.
 One particularly severe shortcoming of ODF is the lack of an open or ISO/IEC-approved object linking and embedding model. Microsoft OLE (the ODF spec references the 1995 work Inside OLE by Kraig Brockschmidt) is used in this respect. In late 2013 OASIS submitted a Referencing Explanatory Report (RER) about using this old work as a reference for ODF. Use of OLE will remain potentially problematic for interoperability until the OLE data structure spec (MS-OLEDS) is listed in the Open Specification Promise.