What is the dominant hemisphere in the first place? I found the following definition (source: Medical Dictionary):
[The] dominant hemisphere [is] [t]he left half of the brain in almost all right-handed people and 85% of left-handed people. This is the hemisphere concerned with language and logical thought and containing the motor areas for voluntary use of the right side of the body. In 15% of left-handed people, the right hemisphere is dominant and subserves speech.
Now, your ducks facing left or right hardly seem to reflect either language, logic, motor lateralization or handedness. Instead, often questionnaires or inventories are used, for example to tease out whether a person is logical, rational and verbal (left-sided); or emotional, manipulative/spatial and creative (right-sided). However, the meaning, relevance, and possible use of hemispheric dominance is debated.
Note that brain lateralization is certainly true for a number of functions, including speech and motor control, but the more dramatic examples of brain lateralization should be interpreted with caution, and I quote from Corballis (1980):
Since ancient times, right and left have been associated with different fundamental classifications, including male and female, good and evil, day
and night, straight and crooked. [...] Interpretations of cerebral asymmetry that emphasize a fundamental duality in cognitive processing between the two sides of the brain, or that locate consciousness in the left side only, are probably modern manifestations of the age-old mythology of left and right.
- Corballis, American Psychologist (1980); 35(3): 284-95