Being 2/3 of "No Hands, No Eyes, No Brain", Handy and Lumpy’s friendship seems to be neutral. While they get along most of the time, and are often seen working together, both of them have behaved in an antagonistic way towards the other at least once.
For the most part, Lumpy seems to like Handy. They are seen working together in several episodes, such as Concrete Solution and In a Jam, at a construction site and in a band respectively. Lumpy also seems pleased to see him in The Way You Make Me Wheel and thanks him for fixing his car's tire. They also work together to survive on the island in Happy Trails Pt. 2: Jumping the Shark, and team up to kill Flaky alongside Sniffles and Flippy. On a more professional note, Lumpy treats Handy as his patient in I Nub You.
This friendship is far from perfect, however. The most famous example of conflict and antagonism on Lumpy's behalf occurrs in Don't Yank My Chain, where Lumpy arrests and beats up Handy, and chains him to The Mole. Lumpy also intentionally murders Handy in Dunce Upon a Time, though he is depicted as a giant in the episode. Lumpy also makes no attempt to help Handy and even steals his heart in A Change of Heart, so he and Giggles can save Disco Bear.
Handy also seems to like Lumpy most of the time. They work together in the aformentioned episodes, and Handy greets Lumpy and fixes his car tire in The Way You Make Me Wheel. They also work together on the island in Happy Trails Pt. 2: Jumping the Shark though this ends when Handy, along with Flippy and Sniffles, throws Lumpy out of the rocket. Handy also goes to Lumpy's theme park in The Wrong Side of the Tracks. He also seems to trust Lumpy to help him build Giggles' new home in Home Is Where the Hurt Is.
Again, however, Handy also is not completely fond of Lumpy. He is obviously annoyed with Lumpy for arresting him and The Mole in Don't Yank My Chain, and both Handy and The Mole attempt to escape. In Happy Trails Pt. 2: Jumping the Shark Handy shows no issue with throwing Lumpy out of the rocket to make it lighter. Handy is also annoyed with Lumpy stitching his and Petunia's bodies together in I Nub You.
- With The Mole, they make up No Hands, No Eyes, No Brain.