Pass Exceptions from rest server to client code?

We are using Spring's RestTemplate and I would like to learn if there's any way to pass a custom exception to the client ... more than just error codes.

I would like to see if I can add checked exceptions from the service layer all the way up to the web controller layer.

Here's how we call a rest endpoint now:

return getRestTemplate().exchange(createUrl, HttpMethod.POST, new HttpEntity(request, getRequestHeaders()),
            OurCustomResponse.class).getBody();

Maybe a dumb question? Thanks all.