Seeing a Blender 2.66 screenshot in 2020 is neat! I approve of your choice to use the version recorded in the course.
Uh oh, did I fail to mention X-symmetry and the symmetrize button in the video? My apologies. By default (Blender factory settings) X-symmetry should be enabled in Sculpt mode. Pretty sure it’s been that way as long as I’ve been using Blender (still true with 2.9)
Thanks for posting this for others to see! No doubt more will be in the same situation wondering about symmetry.