Here is the full list of all 28 Lovers in a Dangerous Spacetime achievements.
Pirouette 15 times in a single level.
Earn one of each type of ship badge.
Defeat Ursa Major without taking a punch.
Beat any level in two player mode.
Rescue every captured friend.
Survive after being fully enveloped by a bomb trap.
Defeat King Cepheus.
Escape a white dwarf without shooting or ramming it.
Finish a level without crossing paths with your partner.
Defeat Orion.
Defeat Cetus.
Fill up all gem slots in a fully upgraded ship.
Defeat Ursa Major.
Beat any level with the help of your trusty space-pet.
Finish a level after removing all fog of war.
Kill 20 enemies with security guns in a single level.
Throw a gem into a station that is being used by another player.
Use every station in a single level.
Finish the tutorial.
Finish a level without taking damage.
Play through a campaign without using any gems.
Defeat Cetus while its head is underwater.
Deliver the final blow to Orion using a solar flare.
Remove all of King Cepheus's shields.
Have 10 unused gems in your ship at one time.
Use three different types of Yamato weapons in a boss fight.
Have 20 cells at one time on the Metal-Beam shield.
Make two Power-Metal missiles collide.