The Shikoku Railway Culture Museum sits beside Iyo-Saijo Station and is exactly the right size for a travel-day stop. Its collection includes early Shinkansen and diesel rolling stock, plus exhibits on Shinji Sogo, the Saijo-born railway executive remembered as the father of the bullet train.
It is not a huge museum, but it is specific and satisfying. Train lovers should go; everyone else can still enjoy the old cars before catching the next limited express.