Q: Why do right-handed people put things in their left pocket, and vice versa?
A: I suppose that the left pocket means the left shirt-pocket because, in the case of pant-pockets, it is the right pant-pocket that most right-handed people usually put things (a wallet, a mobile phone, etc.) in. The reasons that they use the right pant-pocket are probably that it is more convenient and faster and that it consumes less energy than using the left pant-pocket.
The reason that right-handed people usually put things in their left shirt-pocket and vice versa for the left-handed people is probably the same. For right-handed people, putting things in the right shirt-pocket requires more flexing of the finger, wrist, and elbow joints, which is less comfortable and consumes more energy (for muscle contractions). But, unlike your case, I observe that most people who put things in their left shirt-pocket with their right hand will remove the object from the pocket with their right hand, not with their left hand and then transfer it to the right hand, which will take more time and expend more energy.
Finding ways to spend least energy for any action seems to be a well-preserved selected trait in all animals. It certainly gives a basic advantage for survival to the animals. That’s why, mostly, a man or an animal will always try to walk/fly/swim the shortest path it can to the goal. This preference applies to all other actions. This finding ways to spend least energy is called the principle of least effort:
“The principle of least effort is a broad theory that covers diverse fields from evolutionary biology to webpage design. It postulates that animals, people, even well-designed machines will naturally choose the path of least resistance or "effort".”[Ref 1]
“Summary. The observations in this section can be summarized by proposing three fundamental principles underlying the study of movement economy.
The first derives from Hull (1943), to the effect that, when the organism is repeatedly presented with a number of equally reinforced alternative responses, that response requiring the least metabolic energy expenditure will become the dominant response.
The second principle is that, when either the choice of response has been made or no choice given and a goal response specified, over trials the organism minimizes the metabolic energy expended in achieving the task goal. This economy-learning, or efficiency-learning, relationship, as shown in Figure 7, was first suggested by Freeman (1948), but Brener and Mitchell (1989) have also recognized the same principle.
The third principle serves to explain these phenomena - the principle of least metabolic energy expenditure. In being receptive to sensory information about metabolic energy expenditure, organisms select the least effortful coordination and control function, and, with practice, the selected control parameters are refined to attain the task goal with less metabolic energy expenditure. The following section examines the contribution of various sources of informational support for response selection and metabolic energy minimization.” [Ref 2]
Thus, the reason that right-handed people put things in their left shirt-pocket is probably an evolutionary biological reason, not a psychological reason.
It should be noted that evolution works in general but not in every case. And variation is another constant facet of evolution. So, not every selected evolutionary trait will appear in all members of a species or will win out over other competing traits and manifest itself every time in a similar situation. Sometimes, another trait wins out. Therefore, some people or some people sometimes will not behave according to this principle of least effort.
Wikipedia. The principle of least effort
Sparrow WA, Newell KM. Metabolic energy expenditure and the regulation of movement economy. Psychonomic Bulletin & Review. 1998; 5: 173. https://doi.org/10.3758/BF03212943