To start, a clarification of terms: for me, spirituality is a perspective that looks for connectedness, and takes for granted that nothing exists in isolation. This is the basis of compassion, which is arguably our subjective experience of connectedness, and the whole point of spirituality.
When you’re attuned to that connectedness, the state of your surroundings matters. You don’t see yourself as a being in a bubble, unaffected by what’s around you. It’s not that you want to keep your surroundings orderly so that you can be more spiritual, though. It’s more that—strange as this may sound—your compassion moves you to want the best for everything.
That’s hard to explain, but I experience it like this: spaces (or things or people or animals) that show signs of neglect and disarray look to me like they’re suffering. Even though it may not make any sense to think of a neglected room or car as “suffering”, that’s how it feels from a perspective of connectedness/compassion. It’s that sense of suffering that moves me to want to do what I reasonably can to alleviate it. To be “spiritual” is to be responsive to that urge.
In the Zen world, there’s a maxim that if you walk into a temple and see dirt and disarray, then something’s amiss in the spiritual life at that temple. That’s not because Zen temples are “supposed to be” neat and clean. It’s because a community of compassionate people wouldn’t abide suffering like that.