When you appear for H-4 visa stamping and if the officer doesn’t cancel your H-1 visa stamp, then you will have both H-1 and H-4 visa stamps in the passport.
Irrespective of whether your H-1 visa stamp is canceled or not, you can travel to US on H-4 visa. Once in US, an employer can file for COS (change of status) from H-4 to H-1. If its the original H-1 employer, then this will be just the COS; else it will be a cap-exempt petition (aka H-1 transfer) and COS.
Once COS is approved, your status will become H-1 and you can start working for that employer.