There are a number of different ways to rotate photos in Lightroom. What options you have available will depend on whether you're in the Library module or the Develop module.
Rotating Photos in the Library Module
In general, the rotation options in the Library module are quick and simple. They rotate in 90-degree increments either clockwise or counterclockwise. If you want more precise control or need more or less than 90-degree increments then you'll want to switch to the Develop module and use the more fine-grained tools available there.
As with many functions in Lightroom, there are multiple ways to invoke a function. All three of these variations accomplish the same thing. Not every function is available in the Library or Develop module. In this section I'm focusing on the functions available in the Library module. See below for the Develop module.
Toolbar. If you're using the single photo view in the Library view, there are rotate left and rotate right buttons on the toolbar under the image. If you can't see the toolbar, you toggle it by pressing T or go to View > Show Toolbar. The rotation buttons are only visible in the Library module--you won't see them if you're in the Develop module.
Keyboard Shortcut. You can also use a keyboard shortcut. On Mac it's ⌘+[ to rotate left or counter-clockwise and ⌘+] to rotate right or clockwise. The Windows equivalents are Ctrl+[ and Ctrl+].
Main Menu. From the main menu it's Photo > Rotate Left (CCW) and Photo > Rotate Right (CW).
Right Click. You can also access the menu version by right-clicking (or Ctrl-clicking on Mac if you don't have right-clicking enabled), where you'll get the same Rotate Left (CCW) and Rotate Right (CW) options.
Rotating a Single Photo by 90 Degrees in the Library Module
These four ways to access the 90-degree rotation will work in the single-image view in the Lightroom module and apply only to the visible image.
Using the buttons is pretty self-explanatory. One rotates 90 degrees clockwise; the other rotates 90 degrees counter-clockwise. Pressing it again will take it another 90 degrees. If you need to rotate in something other than 90 degrees, you'll need to switch to the Develop module.
Rotating Multiple Photos by 90 Degrees in the Library Module
You can also rotate multiple photos in the Library module by using the same tools. In the thumbnail view, select the photos you want to rotate and rotate using one of the four options above. It applies the same rotation to all the selected photos regardless of their original orientation.
Here's the result with a single counter-clockwise rotation.
Rotate a Photo by 90 Degrees in the Develop Module
If you're already in the Develop module and want to switch the orientation of the photo from landscape to portrait (horizontal to vertical) or vice versa, you can, but there are some slight differences in how you do it.
As you can see, the rotation buttons aren't available in the toolbar anymore. You can still use the same keyboard shortcuts (⌘+[ and ⌘+] or Ctrl+[ and Ctrl+]). You can still use the main menu via Photo > Rotate Left (CCW) and Photo > Right Right (CW). And you can still use the right-click menu, but the rotation options are now grouped under the Transform menu item.
Rotate a Photo Less than 90 Degrees
The 90-degree rotation is a simple way to switch photos from portrait to landscape orientation or vice-versa. But if you're looking for my fine-grained rotation you'll need to use different tools that are available in the Develop module.
The rotation options are under Crop & Straighten tool (which is also what you can use as part of resizing photos, and I also have more details on straightening here). You can access it by click on the crop tool icon, going to menu item Tools > Crop, or just press R.
(There's also a rotation slider under Lens Corrections > Manual, but that's really only useful if you're complementing the other lens correction transformation sliders.)
When using the Crop tool there are different ways to control the rotation.
Straighten a Photo with the Slider
The slider starts by default in the middle. Drag the marker to the left to rotate counter-clockwise and drag to the right to rotate clockwise. You'll see the degree reading to the right of the slider reflect the change measured in degrees.
If you want to reset back to zero you can drag the slider back to the middle. But an even quicker way is to double-click on the word Angle to the right of the slider (this reset trick works on other tools in Lightroom as well).
You can also manually input the amount of degrees you want to rotate. Click on the degree reading and simply type the rotation. Positive numbers rotate clockwise. To rotate counterclockwise, just ad a minus sign in front. As soon as you hit enter or click outside the box you'll see the rotation reflected on the image.
Rotating with the Crop Handles
Once you click on the Crop tool the crop overlay will become visible. It has 8 handle spots--one on each corner and one in the middle of each side. If you click and hold on any of those the cursor will change. If you're right over the crop handle it'll be a cursor that has a straight double-ended line to indicate you're cropping. If you move the cursor slightly outside the image it will change to a bent line indication rotation.
If you click and drag you can then rotate the image.
If you're rotating the photo in order to straighten a horizon, there are two ways to do it.
One of the small features that has quietly crept into the latest versions of Lightroom without fuss is a new Auto straighten option. It's under the same Crop tool function.
With that, Lightroom tries to find a line that looks like it should be a horizon. Your mileage will vary from photo to photo. On some it works well, but it's a pretty blunt tool and only works if there appears to be a clear horizontal line.
A much more precise way to do it is to use the spirit level tool that's in the Crop & Straighten section. Once you click on that tool it will drag the tool out of its storage area.
You then click on a section of the photo and hold down, then drag to another section and release. You'll see a line that you can align with a horizon or even vertical lines or just use to eyeball the rotation. If you need to make further refinements you can still use the slider.
How to Flip a Photo Horizontally or Vertically
If you want do make a mirror image of your photo, rotation isn't going to get you there. What you need is to flip the photo, either vertically or horizontally. What you want is the Photo > Flip Horizontal or Photo > Flip Vertical items from the main menu.
These will make a mirror image on whichever plane you choose.