To make the terriyaki sauce, combine all the ingredients except the cornstarch. Reserve half of the sauce, marinate the squid in the other half for 1 hour.
Prepare the fire. Skewer squid pieces and tentacles. Heat the reserved terriyaki sauce in a saucepan. When it begins to boil, add the cornstarch dissolved in a little water. As soon as the sauce thickens, remove from heat. As it cooks you will baste the squid with this terriyaki sauce.
Grill the squid until it turns dark brown, 2 to 3 minutes per side, depending on the heat, and then turn it over. Baste one last time and then remove skewers from the fire. Serve over rice accompanied with any remaining sauce.