ssh -X authentication with sudo

[Dave Howorth <debian-user@howorth.org.uk>]

I have a machine running debian that I access using ssh. I use the -X
with ssh and can successfully run e.g. xeyes on the debian machine
showing the display on my local machine. But now I want to run a
graphical program that needs to run as root on the debian machine
while displaying on my local machine.

I haven't figured out the correct incantation to use, nor which machine
to run the incantation on :( I tried completely disabling access
control on both machines (xhost +) but still got an error:

$ xeyes
[works with local display]
$ sudo linssid
X11 connection rejected because of wrong authentication.
QXcbConnection: Could not connect to display localhost:10.0
Aborted
$