When and how can I propagate a glorybower? There are suckers coming up at the base of the plant.
Regarding propagating Clerodendrum trichotomum (harlequin glorybower), the book Trees and Shrubs Hardy in the British Isles Vol. I (by W. J. Bean, 1981, p. 667) says that shrubs can be propagated by root-cuttings, or by the young suckers which frequently spring from the roots.
Since you mentioned that there were suckers (gardening term) coming up around the plant, it is most likely the species is C. trichotomum and not C. bungei (one of the others commonly grown in our climate). C. bungei, according to the same source, should be divided in the spring.
Another source, Flora, Vol. 1, (chief consultant, Sean Hogan, 2003, p. 393) says regarding Clerodendrum (genus-level information) that propagation is done by sowing seed in spring or by taking cuttings of half-hardened wood during winter or summer.