You may feel like a salmon swimming upstream for a good portion of the walk but it is certainly possible. I am currently walking from Vercelli to Rome on the Via Francigena. In the fall or next spring, we will likely walk from Rome to Brindisi. The current walk is very well signposted towards Rome and I've seen some signs for the opposite direction as well. Despite the signs, I also recommend Maps.me with the route GPS points. Also, a local phone for calling ahead for a bed is close to mandatory if you want to stay in ostellos. There are not enough pilgrims to have full time hospitaleros.
I've also walked from Navarette to Montserrat / Manresa near Barcelona as part of the Camino Ignaciano. If you choose that route, which is the opposite direction of a few Santiago Caminos, it is also well signposted and supported. It is a challenging but beautiful section, particularly through Los Monegros.
After we reach Brindisi, the next walk is planned to connect Vercelli to Manresa which is supposed to be well signposted mostly through GR routes but that planning is well in the future. Buen Camino!
