Word or phrase for wanting hidden parts of construction to be orderly

What you are describing is perfectionism.

Definition of perfectionism from ODO
noun
[mass noun]
- refusal to accept any standard short of perfection.

Definition of perfection from ODO
noun
[mass noun]
- the state or quality of being perfect
- a person or thing considered to be perfect
- the action or process of improving something until it is faultless

The word I would suggest is meticulous. According to Collins:

meticulous: extremely or excessively careful about details; scrupulous or finicky

Wordnik lists this definition:

meticulous: characterized by very precise, conscientious attention to details.

Although this word doesn't specifically differentiate between the seen and unseen parts of a craft or creation, I do think the attention to detail you described is typically performed by a meticulous person (at least, meticulous in that endeavor).