Why can't I permanently delete photos from my iPhone?

I've deleted a number of images from iPhoto on my iPhone and my iMac but they won't go away. Both devices have the latest OS as of just before I write this.

I do not share an album between the phone and the Mac. I do not have photos in the iCloud. To get them from my phone to the Mac; I import when I connect them. I have emptied the Recently Deleted folder on the iPhone. I went to General, Storage but nothing there.

The pictures are still on my iPhone. I can select but am given no options to delete or hide them. And I don't want to just hide them. I want them to be deleted from my phone.

How do I delete these images?

Thanks.

[Re-Titled by Moderator]