Here is the full list of all 25 Giana Sisters: Twisted Dreams achievements.
Complete a Dream Rush race
Land on an owl while twirling
Collect a yellow gem while dashing
Collect a red gem while twirling
Unlock the power to warp your dreams
Defeat Hansel and Gretel
Defeat Octor Freud
Defeat Gurglewocky
Jump or dash onto three owls without touching the ground
Bounce off a falling turtle
Use a spiky box like a trampoline
Avoid spikes by ducking
Collect all gems in the game
Obtain five stars in all levels
Unlock Hardcore Mode
Unlock Über Hardcore Mode
Obtain a trophy in all Time Attack levels
Obtain a trophy in all Score Attack levels
Complete the first world in Über Hardcore Mode
Complete the second world in Über Hardcore Mode
Complete the game in Über Hardcore Mode
Unlock Cosmonaut Comrade from the SpeedRunners game by completing a race in three different Dream Rush levels
Have a human win a race against bots in a Dream Rush level
Have a human win a race against bots in two different Dream Rush levels
Have a human win a race against bots in three different Dream Rush levels