Snantonu...
The assumption and general practice, within a geodatabase, is that the polygons are converted to polylines in sequential order as they exist in the table. The polylines would begin with the same sequential construction. If the destination was to a shapefile, for instance, or there was some concern that the object id's wouldn't then be matched, then constructing a Long field with a sequential numbering would only help if the attributes were being carried over to the destination polyline... apparently, the attributes aren't being carried over in any event.
To be extra stafe...
- one could perform the polygon to polyline conversion
- convert the polygons to centroids
- spatially join the centroids to the polyline
- save the result
Other approaches could include intersects etc. but I wouldn't even go there until I had finished validating the data (ie compare polygon and polyline perimeters, compare true centroids and a variety of other data validation tests.)