Long time since I've needed to worry about this but there is a certain similarity with RADAR headings. When a controller gives a RADAR heading, that's the one he (or she) expects the "paint" to make on the PPI. It isn't necessarily the one the driver should be seeing on the DI. That's one reason why the fact it is a RADAR heading is stressed. As a tower controller handling a missed approach, "climb straight ahead" expected just that: along the extended centreline.