/usr/include/mutter/meta/ is in libmutter-dev 3.10.4-0ubuntu2.
This file is owned by root:root, with mode 0o755.
..
/usr/include/mutter/meta/atomnames.h
/usr/include/mutter/meta/barrier.h
/usr/include/mutter/meta/boxes.h
/usr/include/mutter/meta/common.h
/usr/include/mutter/meta/compositor-mutter.h
/usr/include/mutter/meta/compositor.h
/usr/include/mutter/meta/display.h
/usr/include/mutter/meta/errors.h
/usr/include/mutter/meta/gradient.h
/usr/include/mutter/meta/group.h
/usr/include/mutter/meta/keybindings.h
/usr/include/mutter/meta/main.h
/usr/include/mutter/meta/meta-background-actor.h
/usr/include/mutter/meta/meta-background-group.h
/usr/include/mutter/meta/meta-background.h
/usr/include/mutter/meta/meta-cursor-tracker.h
/usr/include/mutter/meta/meta-idle-monitor.h
/usr/include/mutter/meta/meta-plugin.h
/usr/include/mutter/meta/meta-shadow-factory.h
/usr/include/mutter/meta/meta-shaped-texture.h
/usr/include/mutter/meta/meta-window-actor.h
/usr/include/mutter/meta/prefs.h
/usr/include/mutter/meta/screen.h
/usr/include/mutter/meta/theme.h
/usr/include/mutter/meta/types.h
/usr/include/mutter/meta/util.h
/usr/include/mutter/meta/window.h
/usr/include/mutter/meta/workspace.h