REACH and CLP
CLP covers most aspects of communicating the hazards of chemicals to end users, except for Safety Data Sheets, which are covered under the REACH regulation.
This means that CLP and REACH interact together to ensure that CLP data is included in REACH SDSs.
However, the data generated from substances which are registered for REACH is used to produce up-to-date CLP classifications (which are agreed by all the co-registrants), so REACH is also a resource for anyone classifying substances for CLP.
If you have registered a substance for REACH, you are legally obliged to use the same CLP-GHS classification as given in the REACH dossier (there can be more than one to choose from, depending on impurities). You are also obliged to ensure that the data in your Safety Data Sheet agrees with the data in the REACH dossier. So you do not have to classify your substance yourself, as this work has been carried out as part of the REACH registration process. (In some cases, if you have other data which gives a different classification, you may carry out an opt-out of part of the registration, in which case you may have to self-classify).
If you have not registered a substance for REACH, you can still view the CLP-GHS classification in the REACH dossier (if it has been registered) and choose whether to use this classification or not. You can also compare this classification to other published sources, such as those listed in the Classification and Labelling Inventory.
However, if you choose to do this, you should be careful about not including data in your SDS from the REACH dossier, as you have not paid to have this data generated, and you could be considered in breach of copyright. It may be reasonable to use textbook data reproduced in a REACH dossier, but if you are in any doubt you should seek legal advice.
Of course, you should not try to register for REACH using the data already published by the Lead Registrant, as this would definitely be in breach of their copyright (unless the data is out of copyright for any reason).