"Respaldo" can mean a computer backup copy, at least in certain countries, but I think the likeliest meaning here is "Vuelta del papel o de un escrito, en que se anota algo" (DRAE, definition 3). In other words, information can be manually recorded on the "carné", the health card (presumably a paper card, on which you can write, rather than a credit-card style plastic one such as we now have in Spain), meaning on the front, and you can also use the back, the "respaldo", of the card. It may seem redundant to specify this, but I don't find it implausible, since it's quite common in official documentation for there to be rules about whether or not things can be written on one side or both sides.

Here's an example of this use of "respaldo", obviously in a different context, from Colombia:

"presentar el original y entregar copia del carné de identificación de la Policía Nacional, el cual al respaldo indica que es el carné de sanidad y social"
http://docplayer.es/1974066-Manual-del-usuario-cliente-saten...