From what I have seen on most shock replacements, the cup is positioned like yours. With that said though, I have seen many with flat washers instead of cupped washers. If you are thinking that the cupped washer is causing the noise, simply replace it with flat washer and see if you still have the noise.

 

Possibly have your sway bar links checked for wear.

 

New sway bar and links were installed with no help in noise. After looking over many videos civic pretty much have same set up in rear for this generation.

All show the washer facing this way. Also the upper mount has drain hole it faces outboard. I will swap the hardware around.

 

All the noise is fixed. 2 sets of bad rear shocks, third set fixed the rear and heavy duty washers for front end sway bar links. Finally all the clanking gone.

 

My guess would have been either bad new shocks or weak old springs. Possibly both. Anyway, glad you got it fixed. Thanks for coming back and posting your fix.