You can reorder app icons on your iPhone's home screen however you want, but Apple does not technically let you place them wherever you want. That means no blank spaces in-between apps. You can have an entirely blank page, and that's about it without jailbreaking—but not anymore.
Thanks to developers William Stoy, Gabriel Whaley, and Eric Aragon, true home screen customization is no longer exclusive to jailbreakers. Through the use of their simple web application, anyone with an iPhone 5 or higher can customize the layout of their home screen with just a few simple steps.
The service itself is called MAKEOVR, so head over to Makovr.io using Safari on your iPhone to get the process started. Again, this will only work on an iPhone 5 or higher running iOS 8—iPad and iPod touch are not supported.
The website provides easy-to-follow instructions, where the first real step requires you to either choose a MAKEOVR wallpaper or to use your own. Depending on which option you choose, the process is slightly different.
If using your own image, apply it to your home screen, tap and hold on any app to enter edit mode, swipe over to a new, blank home screen, and take a screen shot of the blank page (Sleep/Wake + Home buttons).
For a unique MAKEOVR wallpaper, grab it from the website, save it, and apply it to your home screen with Perspective Zoom set off.
Once you upload your image, the application will do its magic and essentially create cutouts of your wallpaper that work as invisible placeholder icons. These are the pieces you use to create spaces in-between apps and reformat your layout.
To export a particular piece, tap on it until it's outlined in red, then tap the Share icon and choose the "Add to Home Screen" option. It will then ask you to label the icon; skip that part and select "Done."
Strategically place the placeholder icon wherever you need it to create the effect you're going for. You can get as creative as you want.
To remove the placeholder icons, just delete them like you would an app. To me, this is much easier to use than the outdated iEmpty tool, which was pretty clunky. For more non-jailbreak hacks for your home screen, see how to make your first home screen page entirely blank, nest folders within folders, and remove stock apps.