Well, to me the first thing that comes to mind is chemical compounds comprised of hydrocarbon chains and rings : ) ahhh , O-Chem... yeah I'm a lab rat, sue me.
Then there's the USDA rules giving time limits for how long a piece of farmland has to sit between the last pesticide application and getting certified (which I *think* is like 50 years but could be wrong, also soil testing for residuals is required), among other things. A farmer can grow organically but not be eligible for a certification yet. Depending on breakdown time from last chemical application, I would often think of the product as organically grown.
To me, I consider something organic if it's fed and protected using substances extracted from an animal (bone/blood/fish meal), vegetable (kelp, horsetail) or mineral (fulvic acid, limestone) rather than chemicals made in a reactor.
Inorganic compounds in fertilizers, such as (using the chemistry definition here) calcium phosphate or potassium nitrate) exist in nature and i would happily give a pass on for something i was told is "organic".
The stabilizers and surfactants in fertilizers and sprays, though, are sometimes organic (again, chem definition) compounds and I would not consider them acceptable for organic growing.
So, to recap: inorganic chemicals can totally be organic.
Organic chemicals aren't usually organic.
Confused yet? It gets better!
Pyrethrin occurs naturally in members of the chrysanthemum family. It's an organic organic!!! The usual activator for it (rotenone) is an organic chemical but not organic grow approved. Neem oil is a cocktail of organic hydrocarbons. And it's suitable for organic growing!
Organic gardening styles should focus on building a harmonious balance in the soil, water, and environment using natural substances plus the bacteria, fungi, and even beneficial insects to help the plants thrive in the world you've built for them : )
Edit: added paragraphs. The formatting shamers on Reddit ( r/talesfromthefrontdesk is my guilty pleasure to read daily) are actually on to something lol