Once the hp_drivers_whatever.run thing had been run. The only problem was the scanner/printer only worked for root, not the ordinary user, lugo. Even though lugo belonged to the group 'scanner'.

Fiddled with udev thus:

Changed /etc/udev/rules.d/z60_libsane.rules. Put in the bit between double #'s:
<snip>
# Hewlett-Packard ScanJet 3500C
SYSFS{idVendor}=="03f0", SYSFS{idProduct}=="2205", MODE="0664", GROUP="scanner", ENV{libsane_matched}="yes"
# Hewlett-Packard ScanJet 3970c
SYSFS{idVendor}=="03f0", SYSFS{idProduct}=="2305", MODE="0664", GROUP="scanner", ENV{libsane_matched}="yes"
#
#
#lugo Put in next two lines to try to get the scanner device owned root:scanner, so lugo may use it.
# Hewlett-Packard ScanJet F2280 All-in-One
SYSFS{idVendor}=="03f0", SYSFS{idProduct}=="2404", MODE="0664", GROUP="scanner", ENV{libsane_matched}="yes"
#
# Hewlett-Packard ScanJet 4070 Photosmart
SYSFS{idVendor}=="03f0", SYSFS{idProduct}=="2405", MODE="0664", GROUP="scanner", ENV{libsane_matched}="yes"
# Hewlett-Packard ScanJet 3770
SYSFS{idVendor}=="03f0", SYSFS{idProduct}=="2505", MODE="0664", GROUP="scanner", ENV{libsane_matched}="yes"
# Hewlett-Packard ScanJet 3800
<snip>

non-USB. separate ports for mic & headphone. Works great! No issues at all.