I've done it, my picture looks normal at last! Because my "image edition software" doesn't let me manipulate with EXIF, auto-rotation and stuff like that (I use modest Microsoft Offise Picture Manager, find it quite sufficient), I googled "EXIF edition", selected Free Image and Photo Viewer (
www.exifsoftware.com), downloaded it, opened problematic picture, and it was turned at 90 degree indeed, with orientation tag in EXIF reading "rotated 90 right". I rotated it back and saved it. Then uploaded picture at Panoramio - excellent, picture is orientated as it must be. However EXIF of the picture disappeared, I cannot see it on Free Image and Photo Viewer anyway. I don't know if it is a problem, have never used Exif before and have no idea whether behavior of a picture with EXIF and one without it is the same. Hope it is.
Matthew, thank you for help!
My best regards, Elena.