Upon entering the Inverted Castle, you'll find yourself in the Inverted Castle Entrance. All areas are mirrored and have new names (e.g., Anti-Chapel, Reverse Caverns). The enemies are much tougher, often being powered-up versions of their normal castle counterparts. Your primary objective is to explore each area, find the five Dracula Relics, and activate all Save Rooms and Teleporters.
The relics are: Tooth of Vlad, Rib of Vlad, Heart of Vlad, Eye of Vlad, and Ring of Vlad. Each is typically found after defeating a boss or in a hidden area. Use your map extensively and don't be afraid to backtrack. Many areas will require advanced use of your Bat, Wolf, and Mist forms. Prioritize leveling up and finding better equipment.