In the vast realm of home maintenance, a striking combination of fluid mechanics, thermodynamics, and heat transfer comes to the fore in the form of HVAC system upkeep. It is a task that demands acute attention, yet often falls into the abyss of neglect. While the decision to hire professional HVAC cleaning services may seem straightforward, it is, in fact, enveloped in layers of complexity that can only be unraveled by a discerning eye. Here, we will dissect six aspects of HVAC cleaning services that I wish I'd been privy to prior to engaging a professional in this field.

Firstly, the significance of the maintenance of air ducts and vents cannot be overstated. The reason is rooted in the basic principles of Epidemiology. An HVAC system, when poorly maintained, can become a breeding ground for allergens, mold, and other pathogens, thereby exacerbating the risk of respiratory conditions. Regular, professional cleaning curbs this risk, maintaining indoor air quality and indirectly impacting the health of inhabitants.

Secondly, the HVAC cleaning industry is replete with its own share of charlatans. It mirrors the paradox of choice, a concept in behavioural economics where an excess of options leads to information overload and suboptimal decisions. Therefore, it's pivotal to validate the credentials of the service in question. The gold standard in this regard is certification from National Air Duct Cleaners Association (NADCA), which mandates adherence to strict standards and ethical guidelines.

The third point to consider is the cost-analysis. This process should accord with the principles of maximising utility, a cornerstone concept in microeconomics. Prices of HVAC cleaning services can vary widely, often ranging from $300 to $500. However, the low end of this spectrum might involve limited cleaning, while top-tier prices should ideally include comprehensive services like coil and blower motor cleaning. Your decision should align with derived benefits and financial constraints.

Fourthly, it's crucial to understand the technology employed in HVAC cleaning. A common method is 'source removal', involving a mechanical agitation to dislodge debris, and a high-powered vacuum to eradicate it from the system. However, advances in the sector have birthed the 'Rotobrush cleaning' method, where a rotating brush scrubs the duct walls while a vacuum sucks in the dislodged contaminants. The former is more rudimentary and less effective than the latter, highlighting how technological evolution can enhance efficiency and optimize results.

Fifthly, the temporal aspect of HVAC cleaning is worth considering. While there is no universal decree, the Environmental Protection Agency recommends professional cleaning every 3-5 years. However, this timeframe is influenced by a myriad of factors including climate, pets, allergies, and the age of the HVAC system. Thus, the application of Bayesian inference, updating the probability estimate as new data arises, might be the best way to determine the cleaning schedule.

The final point revolves around post-cleaning protocols. HVAC cleaning could potentially result in damage to the system if undertaken by inexperienced or ill-equipped professionals. Therefore, a post-cleaning inspection is advised to confirm system integrity. In the words of the great mathematician George Pólya, "It is better to solve one problem five different ways, than to solve five problems one way". This approach can be applied here as well, by cross-verifying the post-cleaning condition of the HVAC system, ensuring the utmost efficiency and longevity.

In conclusion, HVAC cleaning services require a meticulous analysis to ensure optimal value. The interplay of various sciences, from economics to epidemiology, alongside evolving technology and ethical considerations, makes it a fascinating, albeit complex, decision. A wise consumer is aware of the potential pitfalls and equipped with the knowledge to navigate this labyrinth with finesse.