Thanks to the commenters, who pointed me in the right direction, I have now identified the book as Galactic Patrol by E. E. "Doc" Smith. The text is available from Project Gutenberg Canada.
I have misremembered several of the details:
- The main character (Kimball Kinnison) wasn't a detective as such, but one of the Lensmen, a kind of interstellar quasi-military police force. He was however called on in the book to investigate a murder using telepathy (Chapter 19, "Judge, Jury and Executioner").
- As per John Rennie's comment, the sense is indeed called sense of perception rather than absolute perception.
- The other alien is Lensman Tregonsee, a Rigellian. Kinnison does not swap senses with Tregonsee, but is granted the sense of perception later on in the book by the Arisians. Before this happens, both Kinnison and Tregonsee express the desire to have the other's senses:
[Kinnison] "I have often wished that I could have your sense of perception, if only for a day. It must be wonderful indeed to be able to perceive a thing as a whole, inside and out, instead of having vision stopped at its surface, as is ours. And to be independent of light or darkness, never to be lost or in need of instruments; to know definitely where you are in relation to every other object or thing around you--that, I think, is the most marvelous sense in the Universe."
[Tregonsee] "Just as I have wished for sight and hearing, those two remarkable and to us entirely unexplainable senses. I have dreamed, I have studied volumes, on color and sound. Color in art and in nature; sound in music and in the voices of loved ones; but they remain meaningless symbols upon a printed page. However, such thoughts are vain. In all probability neither of us would enjoy the other's equipment if he had it, and this interchange is of no material assistance to you."
The planet referred to is Trenco, which has the dust-storms and vegetation as I remember. Tregonsee is not from Trenco, but is assigned to the Lensman base there:
"I can locate you, but you cannot locate me," came the dry reply. "Everyone knows that Trenco is peculiar, but no one who has never been here can realize even dimly how peculiar it really is. Detectors and spy-rays are useless, electro-magnetics are practically paralyzed, and optical apparatus is distinctly unreliable. You cannot trust your vision here--do not believe anything you see. It used to require days to land a ship at this port, but with our Lenses and my 'sense of perception,' as you call it, it will be a matter of minutes."
Kinnison obeyed; and, released from all duty, the visitors stared in fascinated incredulity into the visiplate. For that at which they stared was and must forever remain impossible of duplication upon Earth, and only in imagination can it be even faintly pictured. Imagine all the fantastic and monstrous creatures of a delirium-tremens vision incarnate and actual. Imagine them being hurled through the air, borne by a dust-laden gale more severe than any the great American dust-bowl or Africa's Sahara Desert ever endured. Imagine this scene as being viewed, not in an ordinary, solid distorting mirror, but in one whose falsely reflecting contours were changing constantly, with no logical or intelligible rhythm, into new and ever more grotesque warps. If imagination has been equal to the task, the resultant is what the visitors tried to see.
The part about tracing a circuit board comes from this quote:
Instead, with his new sense of perception, he sought out the conduits leading to those cells and traced them down, through concrete and steel and masonry, to the control room far below.
