Transfer files between trisquel_10 Thinkpad laptops with crossover ethernet cable

Don't you have a router?

Even if I get the ethernet connection up, what applications give me access to the other computer's desktop ?

Trisquel 9 and 10 ship with Vinagre. Since it is now discontinued in favor of GNOME Connections, it is possible that Trisquel 11 will ship GNOME Connections instead. I do not know. There are other VNC clients.

On the other end, you need a VNC server. There are several too. Trisquel ships with Vino by default. However, here again, that project is now abandoned in favor of GNOME Remote Desktop, which is in Trisquel's repository.

Yes, I do have a router, but the crossover ethernet cable ought to be more secure.
Trouble is, after I create the ethernet connection on the T420 with sudo ip ad add 10.0.0.10/24 dev enp0s25 what I see in the list of connections is enp0s25, analogously to my WiFi connection, but on
the T400, after the similar command sudo ip ad add 10.0.0.20/24 dev enp0s25 there is no network connection named enp0s25. And that unnamed ethernet connection won't stay up for very long.

Vinagre fails to detect the T420's name (george-Thinkpad-T420), address (10.0.0.20) or the named network connection (enp0s25). ping -c3 10.0.0.10 and ping -c3 10.0.0.20both work, but the second one fails if I don't re-up that connection frequently.

"What applications give me access to the other computer's desktop?"

Caja knows how to handle file transfers. Just hit CTRL+L and ssh into the other machine. You'll get a graphical window just like any local folder on a computer within your LAN.

Something like 'ssh://user@remote_host' or 'ssh://user@192.168.1.101'

You should use a router or a switch.

Crossover Ethernet cables are something from mid 90s, most NICs from ~2004 and later should work and connect with each other just fine without need of crossover Ethernet cables thanks to something called Auto MDI-X, if a crossover cable is giving you trouble try using a straight cable instead. If it keeps giving issues, you may try setting the respective IPs of each NIC as default gateways of the other, and run a ping/nmap to see whether they return a response or not and to verify if the SSH port (default TCP 22) is opened, i'd also abstain from using a subnet mask with more than 254 available IPs as it's pointless when connecting only two machines and may require extra steps or cause furtherissues, set the subnet mask to 255.255.255.0.

As for the transferring method itself, i'd just mount the directory in question in one of the machines with SSH over FUSE (sshfs) and use regular copy through command line or your preferred file manager.

https://en.wikipedia.org/wiki/Private_network

Typically network mask for a 10. network is 255.0.0.0 while the 192.168. network would typically be 255.255.255.0 and 172.16.0.0 would be 255.255.0.0.
10.*.*.* is a private address space for huge networks while 192.168.*.* is private for networks where 255 nodes will suffice.

But 10.*.*.* can be subdivided so I don't know if that's your problem.

'What is the gateway for those addresses ? '
Gateway for 10.0.0.10 would typically be 10.0.0.1 (but can be setup differently) and that would be assigned to the local router. I wonder what would happen if you made each laptop the other's gateway.

Or maybe consider disconnecting the router you have from the internet till you get the transferring done so you can use it as a reliable gateway.

re 'there is no network connection named enp0s25'
On my system: sudo ifconfig -a reveals my interface name as enp2s0. So I would go with
'sudo ip ad add 10.0.0.20/24 dev (whatever sudo ifconfig -a reveals on that machine)

Success ! Following the excellent tutorial that I found here:
https://phoenixnap.com/kb/ssh-to-connect-to-remote-server-linux-or-windows
where it's said: How to Use SSH to Connect to a Remote Server in Linux or Windows...
Aside: Don't pay attention to the last two words in the preceding sentence.
Everything (99%) of the syntax and steps work as written. The remaining 1% can be
intuitively worked out from an octogenarian's experience.

I can now log in to the remote machine with ssh george@10.0.0.20 and I can get the contents of a directory there with ls /media/george and I see the remote equivalent of my local /media/george folder.

Next, I'll set up to use Filezilla to make the file transfers; I have extensive experience
with that application because it was always 100% accurate.

When I invoke the network connections withsudo ip ad add 10.0.0.10/24 dev enp0s25andsudo ip ad add 10.0.0.20/24 dev enp0s25
where does the network device enp0s25 reside ? I can see it in each ThinkPad's Network list,
but neither Caja nor Mate's file manager can find that network.

On the other hand, it took me only a minute or two to start FileZilla and connect to the remote machine,
select a file, and copy it to my local machine.

In the interest of learning to use the console, there are tasks that are more efficient with the console's
commands.

After reading loldier's excellent & informative suggestions, I chose to install sshfs and
then tried george@george-ThinkPad-T400:/media/george$ sudo sshfs george@10.0.0.20:/media/george/Data-TP/ /media/george/TP-420/ -o slave where the -o slave option was intended to give me free access though the console and/or file manager, but the command stalled and when I stopped it with control-c the response was read: Interrupted system call.
I renamed the mount point so as not to confuse the mounted remote file system with the
local file systems. I invoked the -o slave option because the mounted remote system would
be unreadable otherwise.

Naturally, as that seems to be my way, I tried the Caja approach and found myself staring
at the complete file system of the remote ThinkPad. That is definitely the best:
Control -L, them ssh://george@10.0.0.20; then P/W one; then P/W two. Presto ! The Caja
file listing appears, just like Mate's file manager. No mounting of remote filesystems;
no hassles with permissions.

Thanks for everyone's patience.

This was easy when both computers had an active network connection between 10.0.0.10
and 10.0.0.20, but I inadvertently disconnected the network and now every attempt to
reconnect with Control -L, then ssh://george@10.0.0.20 times out. It's as through the
system is waiting for passwords, but it's not asking for any. I've even set up the
local network with sudo ip ad add 10.0.0.20/24 dev enp0s25 and sudo ip ad add 10.0.0.10/24 dev enp0s25 The other computer still has an active connection to enp0s25.

Magic Banana suggested:
Try to log in the remote machine, executing in a terminal:
ssh george@10.0.0.20

Today both machines have been restarted, and I tried to start the T420's
end of the network with
sudo ip ad add 10.0.0.20/24 dev enp0s25 which induced the T400 to start its enp0s25 network.
Then I executed the following two commands in the "remote" T420 with
sudo ip ad add 10.0.0.10/24 dev enp0s25 and
sudo ip ad add 10.0.0.20/24 dev enp0s25
but executingssh george@10.0.0.20 in the T400 is greeted with the response:
ssh: connect to host 10.0.0.20 port 22: Connection refused.
Conversely, executing
ssh george@10.0.0.10 in the T400 elicits:
ssh: connect to host 10.0.0.10 port 22: No route to host
At the other end of the network cable, ssh george@10.0.0.10 gets the response:
ssh: connect to host 10.0.0.10 port 22: timed out.
In other words, the T420 at 10.0.0.20 cannot get through to the T400 at 10.0.0.10
the opposite of what was the experience in my previous post.

Previously, one network connection could be made, but not the other.
Yesterday, both ends of the Ethernet cable "connected" and FileZilla worked flawlessly.
Yesterday and today, both network connections made easily with:
T400 ==> sudo ip ad add 10.0.0.20/24 dev enp0s25 Then ssh george@10.0.0.20 ==> creates enp0s25
T400 ==> sudo ip ad add 10.0.0.20/24 dev enp0s25 Then ssh george@10.0.0.20 ==> creates enp0s25
Today: Cannot connect with any application, but enp0s25 is listed in Ethernet Networks
Yesterday: Connected with FileZilla, transferred ca. 400GB (750,000 transfers, zero errors).

Improvement is needed ==>
Select network from a list (enp0s25, like the WiFi connections).
Select application from a list (Caja, FileZilla, Remmina, Remote Desktop Viewer).
Provide credentials (username, password, local address).

Caja is the worst: Asks for password multiple times, popup has zero tolerance for any typing error, no reason for failed connection.