"Organic", real honest to goodness organics, is considered true in the United States when a product bears the USDA Organic Seal; has been certified organic; and contains 95% or more organic ingredients. That said, true organics is a far reaching term that's not always clearly defined and many non-certified organics are in fact, organic too.
"Integrity" as defined by Websters is the "Firm adherence to a code of especially moral or artistic values." Integrity is very much linked to "incorruptibility" as well, meaning, if you stand by an ideal or issue, you stick with it, no matter what.
Generally speaking, if you're immune to corruption you're incapable of having your specific belief or ideals surrounding an issue changed via bribes or other means, thus protecting said ideals from decay or dissolution.
When you smash "organic" and "integrity" together you get an issue that does really mean something important to many consumers and organic advocates in the U.S. Usually this term is used to describe the action of maintaining organic integrity, meaning to uphold organic quality as it has been set fourth by the Organic Foods Production Act, the National Organic Standards Board (NOSB) and the National Organic Program (NOP) in general.
Organic integrity is also often used when there is a lack of it. For example, "XYZ shampoo company lacks organic integrity because they label their shampoo as organic when it really only contains 45% organic ingredients." A company using the term organic to sell a product for more gains is not upholding organic integrity, brings the price of real organics up and worse, makes the entire organic industry look shady. You could also say that this shampoo company is greenwashing, but at the core they lack integrity.
Another example of organic integrity gone astray is if the NOSB attempts to allow new, potentially harmful or questionable ingredients onto the National List, such that the original ideals of organics change. This affects the meaning of organics as a whole and increases consumer confusion and annoyance with organics, thus bringing down organic integrity.