A simple configuration: LAN + direct connect simultaneously [closed]

This should be an easy configuration, but I'm struggling to Google for it; the vocabulary eludes me.

I've got a desktop with onboard gigabit which finds its way to the internet via the router. Naturally.

I've got a server with onboard gigabit which finds its way to the internet in much the same way. No surprises.

The router provides DHCP to the clients.

I grabbed a couple of old Intel 10GbE cards from eBay and am keen to create a 10Gbit direct connection between my desktop and the server. I've got the cards in and the cables run. I assigned IPs to all four ports (two on each card) statically and connected one port on each. When I enable the interfaces, I get a reported 10Gb connection, but cannot figure a means by which to establish communication between the two.

Here's a map, for completeness' sake.

I've touched nothing with regards to routing tables. ip route shows whatever it defaulted to on both machines. On the desktop, for example:

[patrick@manifold-arch ~]$ ip route
default via 192.168.4.1 dev enp4s0 proto dhcp metric 100 
192.168.4.0/22 dev enp4s0 proto kernel scope link src 192.168.4.73 metric 100 
192.168.5.0/24 dev enp9s0f1 proto kernel scope link src 192.168.5.203 metric 101 

And all I've tried to do is ping back and forth as well as SSHing from server to desktop (I've got sshd enabled on my desktop). Ping says the destination is unreachable and SSH says no route to host. I know the topology is nonstandard, but beyond ensuring the physical connection, I'm not sure how to troubleshoot this. Any pointers or direction would be most welcome!

EDIT

I've changed the IP addresses to 172.16.0.x, but the behaviors described above persist.

[patrick@manifold-arch ~]$ ip route
default via 192.168.4.1 dev enp4s0 proto dhcp metric 100 
172.16.0.0/24 dev enp9s0f1 proto kernel scope link src 172.16.0.4 metric 101 
192.168.4.0/22 dev enp4s0 proto kernel scope link src 192.168.4.73 metric 100 

The /22 subnet on the 1Gb interfaces is swallowing up (overlapping) the /24 addresses of the 10Gb interfaces (i.e. packets are sent out the wrong interface). Adjust the addresses on the 10Gb interfaces and you'll be fine. I'd suggest something in the 10.x.x.x range to make it obvious they're "different".