Something that I like to do with my students to introduce mobius strips (before predictions are made) is to bring two pairs of students to the front of the class. Each group gets a scissors. One group gets a regular loop of adding machine tape. The other gets a mobius strip. Make the strips long enough so that the groups can not see the twist. Announce the contest is to see who can cut their loop in half and get two separate loops. The mobius group gets quite a surprise when their loop is cut in half. This then can promote students to think about and predict what will happen when other types of cuts are made.
I have also had students write cyclic stories on a mobius strip so that they continue on and on. For example the a student could start at the beginning of the day, getting up, eating breakfast, going to school, etc, coming home, eating, going to bead, and THEN ending up at the same point on the mobius strip.