By completing certain tasks in the game, you can unlock competitive time trial versions of the levels.

Time Trial	How to unlock
Space Station 1	Rescue Violet and find three trinkets.
Space Station 2	Rescue Vitellary and find twelve trinkets.
The Final Level	Complete the game and find eighteen trinkets.
The Laboratory	Rescue Victoria and find six trinkets.
The Tower	Rescue Vermillion and find nine trinkets.
The Warp Zone	Rescue Verdigris and find fifteen trinkets.