Is it possible to use Linux/unix's dd to make an image backup of an android phone with the phone off? My current position:

Nexus 5
Unlocked with TWRP Recovery

I've been wanting to "rom hop" for a bit, but I'd like to have a good, solid foundation to return to in case of catastrophic failure.

If this were just a normal flash drive, on my computer I'd just 

sudo dd if=/dev/sdX of=~/my.img


However with a cell phone I'm not sure if it will work the same way.

I saw this answer linked while I typed this up, but it looks like the guy in that question specifically wants the /data partition (though he might not have known it at first).

tl;dr: How do I make a dd image of my android phone from my linux laptop?
You can't access any filesystems over USB (or any other way) with the phone turned off. To make a full backup of user data and the ROM (system image), you need to boot into your custom recovery and use that to make a Nandroid backup.