keyboard-shortcuts – Access GNOME Shell Aggregate Menu via Keyboard

Use xmodmap -pm to verify that both keys have the mod4 modifier associated with them.

vikktakkht:686 Z$ xmodmap -pm
xmodmap:  up to 4 keys per modifier, (keycodes in parentheses):

shift       Shift_L (0x32),  Shift_R (0x3e)
lock        Caps_Lock (0x42)
control     Control_L (0x25),  Control_R (0x69)
mod1        Alt_L (0x40),  Alt_R (0x6c),  Meta_L (0xcd)
mod2        Num_Lock (0x4d)
mod3      
mod4        Super_L (0x85),  Super_R (0x86),  Super_L (0xce),  Hyper_L (0xcf)
mod5        ISO_Level3_Shift (0x5c),  Mode_switch (0xcb)

Note that it's not enough that it's named Super_L or Super_R; it's the modifier mapping that matters.

You can use xmodmap to associate keysyms with modifiers, but if your system is at all modern (and not OS X) then it uses Xkb and you probably want to examine your Xkb configuration instead of trying to change it with xmodmap. (Look at setxkbmap and friends, although probably there is a GNOME 3 interface to it that you should use.)

Gnome 3 uses DCONF to store the preferences in a single binary file: ~/.config/dconf/user.
As per the Gnome docs, it is recommended to save only the settings that you need and restore them with either dconf or gsettings. However, gsettings is only able to restore the value(s) for one single key at a time (plus, the value must be quoted) and that makes it a bit awkward for this kind of task. Which leaves us with dconf.
So, in this particular case, save the current settings for gnome-shell keyboard shortcuts¹:

dconf dump /org/gnome/shell/keybindings/ > bkp

Here's a bkp sample:

[/]
toggle-message-tray=['<Super>m']
open-application-menu=['<Super>F1']
toggle-application-view=['<Control>F1']
focus-active-notification=['<Super>n']
toggle-recording=['<Control><Shift><Alt>r']

Load the settings on another system:

dconf load /org/gnome/shell/keybindings/ < bkp

1: WM and Media Keys shortcuts belong to different schemas:

/org/gnome/desktop/wm/keybindings/
/org/gnome/mutter/keybindings/
/org/gnome/mutter/wayland/keybindings/
/org/gnome/settings-daemon/plugins/media-keys/

Note that dconf only dumps non-default values so if you run e.g.

dconf dump /org/gnome/desktop/wm/keybindings/

and don't get any output that means there's no custom WM shortcut defined.

As a side note, dconf-editor is a tool that helps visualizing dconf settings structure, i.e. schema [:path] key value, the type and the default values of any key etc.

For the record, saving the preferences with gsettings:

gsettings list-recursively org.gnome.shell.keybindings > bkp

bkp sample:

org.gnome.shell.keybindings focus-active-notification ['<Super>n']
org.gnome.shell.keybindings open-application-menu ['<Super>F1']
org.gnome.shell.keybindings toggle-application-view ['<Super>a']
org.gnome.shell.keybindings toggle-message-tray ['<Super>m']
org.gnome.shell.keybindings toggle-recording ['<Control><Shift><Alt>r']

Now loading the preferences (as I said, for each line in the backup file you need a separate command and don't forget to quote the values):

gsettings set org.gnome.shell.keybindings focus-active-notification "['<Super>n']"
gsettings set org.gnome.shell.keybindings open-application-menu "['<Super>F1']"
gsettings set org.gnome.shell.keybindings toggle-application-view "['<Super>a']"
gsettings set org.gnome.shell.keybindings toggle-message-tray "['<Super>m']"
gsettings set org.gnome.shell.keybindings toggle-recording "['<Control><Shift><Alt>r']"