SEOUL—At the Consumer Electronics Show in Las Vegas this week, executives from Samsung Electronics Co. and LG Electronics Inc. will talk about ultra-high-definition televisions and mega-capacity refrigerators. But the real battle between the South Korean firms may hinge on the humble washing-machine door.

In recent weeks, a cycle of recriminations between LG and rival Samsung over washing-machine door hinges has spun out of control.

The squabble began in September, when Samsung says that several of its “Crystal Blue” washing machines, which retail for about $2,700 each, were damaged at two Berlin shopping centers.

One of the culprits, according to Samsung: Jo Seong-jin, the 58-year-old president and chief executive of LG’s home appliance division, who Samsung says personally helped to break several of the high-end washing machines.

Hogwash, says LG. If Samsung’s state-of-the-art washing machines are indeed broken—LG says it hasn’t been given access to the appliances in question—it is only because of the shoddy quality of Samsung’s door hinges, the rival company says. Mr. Jo declined to comment on the incident.