What's the reason some of the Detective Neuro's manga cover's page are upside down?
There's no particular reason other than &quot;because the mangaka wanted it that way&quot;.
It's just one of the many omake/additional fun things about the tankobon (according to Japanese Wikipedia):

The characters on the cover page are upside-down on vol. 1-10, as-is on vol. 11-20, and left-turned on vol. 21-22(23)

Some other trivia:

The name of the criminal of that volume

 is hidden in the background of wall-of-text in the table of contents (except vol. 15, 16, and 23), e.g. vol. 11's criminal name. (Source)


The illustration on the middle cover of vol. 10-19

 uses &quot;Hanging Yako&quot; representing &quot;1&quot;, e.g. vol. 18. (Source)


The cover pages for vol. 13-20

 are connected: 13-16 and 17-20. (Source)