I'm considering buying a new HT radio (Hong Ying HX-860UV Handheld) from China. The company says the model is approved by the FCC. Since I can't see it in person (only a photograph), I can't confirm. How can I be sure they are telling the truth?
The FCC provides a good summary of their authorization process on their website. The manufacturer must get the device tested at an FCC Recognized Accredited Testing Laboratory. The results are then filed with FCC, and the device must be appropriately labeled and documented. If you can get a look at that documentation you will find an FCC compliance statement and an FCC ID.
If you can get the FCC ID of the device, then you can search for it in the FCCs database. In addition to demonstrating that the device has been tested and registered with the FCC, you can also see the test results which includes some interesting performance data about the device.
All FCC IDs are prefixed by an FCC Registration Number which indicates the organization making the filing. For well-known companies, it's usually easy enough to find the FRN by a bit of searching. For example, Apple's FRN is BCG.
Unfortunately in your case of a cheap Chinese manufacturer, this information is more difficult to find, especially if it is not in fact registered with the FCC. I did a bit of searching, and while I was able to find a few "Hong Ying" registered companies, none of them seemed to be the manufacturer of your radio. Perhaps with a bit more searching, or by obtaining the relevant documentation, you will be able to find the registration.