I think this is actually a good example of the point I'm trying to make.
There appears to be an underlying mechanism here that's worth understanding that may have real effects. None of this suggests that left nostril 'relaxation' and right nostril 'energy' is a real thing. The closest may be that last link which I don't have access to.
The reason understanding the underlying mechanism is important is because it allows you to differentiate what could be real (potentially switching airflow in nostrils because "the olfactory system maximizes the ability of its distributed receptor neurons to encode differentially absorbing odors.") and what is pseudoscience "you could choose to be either energetic or restful by picking one nostril to breathe out of for awhile.".
The latter sounds suspicious to me and these links don't seem to really support it. I'm extra suspicious when there's a strong interest in motivated reasoning (yoga-affiliated people being bought into it) to find it correct. It doesn't mean it isn't true, but it's less likely to be so.
The cycle can still be a real thing, there may still be real effects, and the left/right relax/energy thing can still be nonsense. That's why understanding what's going on is useful.
There's actually a lot of research out there. Did you try to search for any? It's not scepticism to compare something you cannot believe because you haven't bothered to investigate, with well known falsities such as crystal healing.
Not only does the research support the existence of a nasal cycle where one nostril is dominant at different times, it also supports a different pathway to the brain and a different neurological response. On top of that, psychological research shows links between breathing patterns and emotional response, in both directions, and then there's the wealth of meditative research. This stuff isn't controversial.
From [Taste and Smell]:
> The FN model may also explain the observation by Sobel et al. [36] that, when performing an odor threshold test, humans sniff longer when using the nostril with the lower flow rate. (Subjects usually have different flow rates in their two nostrils because of cyclic changes in the size of their nasal cavities.)
From [Nature]:
> The flow of air is greater into one nostril than into the other because there is a slight turbinate swelling in one. The nostril that takes in more air switches from the left to the right one and back again every few hours
From [Stanford News]:
> Michael Leon, a professor of neurobiology and behavior at the University of California at Irvine said… [the] data suggest, that the olfactory system maximizes the ability of its distributed receptor neurons to encode differentially absorbing odors.
From [Measuring and Characterizing the Human Nasal Cycle]:
> Nasal airflow is greater in one nostril than in the other because of transient asymmetric nasal passage obstruction by erectile tissue. The extent of obstruction alternates across nostrils with periodicity referred to as the nasal cycle. The nasal cycle is related to autonomic arousal and is indicative of asymmetry in brain function. Moreover, alterations in nasal cycle periodicity have been linked to various diseases.
From [Hemispheric lateralization in the processing of odor pleasantness versus odor names]:
> These findings are consistent with previously demonstrated neural laterality in the processing of olfaction, emotion and language, and suggest that a local and functional convergence may exist between olfaction and emotional processing.
From [Respiratory feedback in the generation of emotion]:
> This article reports two studies investigating the relationship between emotional feelings and respiration. In the first study, participants were asked to produce an emotion of either joy, anger, fear or sadness and to describe the breathing pattern that fit best with the generated emotion. Results revealed that breathing patterns reported during voluntary production of emotion were (a) comparable to those objectively recorded in psychophysiological experiments on emotion arousal, (b) consistently similar across individuals, and (c) clearly differentiated among joy, anger, fear, and sadness. A second study used breathing instructions based on Study 1’s results to investigate the impact of the manipulation of respiration on emotional feeling state. A cover story was used so that participants could not guess the actual purpose of the study. This manipulation produced significant emotional feeling states that were differentiated according to the type of breathing pattern. The implications of these findings for emotion theories based on peripheral feedback and for emotion regulation are discussed.
I could go on.
As to meditative practices, there's been extensive research for nigh on 50 years now, and for the past 20 (at least, as far as I'm aware) they've been using fMRI scanners to provide objective results[Oser] on the link between meditation as a practice and mental state.
You just have to look.
[Taste and Smell] https://books.google.com/books?id=fuxS-p6bpuwC&pg=PA12
[Nature] https://www.nature.com/articles/46944
[Measuring and Characterizing the Human Nasal Cycle] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5053491/
[Stanford News] https://news.stanford.edu/news/1999/november10/smell-1110.ht...
[Hemispheric lateralization in the processing of odor pleasantness versus odor names] https://www.researchgate.net/publication/12712833_Hemispheri...
[Respiratory feedback in the generation of emotion] https://www.tandfonline.com/doi/pdf/10.1080/0269993014300039...
[Oser] https://www.lionsroar.com/the-lama-in-the-lab/ I include this as it's a personal favourite of mine and covered more extensively in Daniel Golman's book, Destructive Emotions.