Knowing that every little (and big) part of our life can be a reflection of the work of God in us, we should seek to be dedicated and disciplined in cleaning and tidying. In this, others can see the work of the Spirit in your life.
Does it matter if you clean under the rug? Nobody will see it either way. God will see it, but more important is that He will see your heart and the reason why you did or didn't clean there.
Some people are just obsessive about cleanliness, and this can go way too far. This isn't likely to be a fruit of the Spirit.
Desiring to do a job properly, to show the character of God in you, not showing off but wanting to show off God, a demonstration of character and not words. This is how the Spirit works in cleaning and tidying.