Thanks for your quick replies. It looks like my Spanish Straightener
will work.
You won't be the first person to bring an Spanish appliance in to Switzerland.
Basically any appliance bought in continental Europe that has an type C plug will fit in a Swiss socket. They are designed for that, so you are not even breaking any regulations.
The normal typ C plug has two prongs, and has a flattened hexagonal shape.
If you have a round plug, with two prongs, and both a whole and clips on the side for the safety earth you have a type E/F plug.
Most countries in Europe have either type F or type E sockets, and for convenience most appliances that need an earth connection come with a plug that fits in both.
There is a nice overview here:
https://www.worldstandards.eu/electricity/plugs-and-sockets/
Type C fits in Swiss sockets, type E/F not, so then you would need an adaptor. You will find them in most supermarkets and electronics stores.
We Swiss have the same problem. When I order an appliance over the internet it will often come with a E/F plug, not a J plug, so I often need adaptors as well.