Ok, that is not normal (you can check against a cat if you would like to see what normal is - it should be almost impossible to put the foot upside down in the first place).
I am concerned that your dog may be experiencing back pain due to Intervertebral Disc Disease (IVDD, herniated disc).
IVDD happens when the material in the discs between the boney vertebrae in the back ruptures out and presses on the spinal cord. There is a lot of internal swelling when this occurs, leading to pain and decreased nerve function. This can happen in conjunction with arthritis of the vertebrae, because that causes decreased flexibility between the vertebrae.
Basically, the vertebrae are bones that protect the spinal cord which runs through a hole in the vertebrae. Each vertebrae has a little "shock absorber" between it and the next vertebrae, called a disc. The disc it a lot like a jelly donut! It has a fibrous part (the bread of the donut) and then a squishy substance in the middle (the jelly).
In SOME dogs, this jelly in the middle becomes chalky and hard as they age. Dachshunds are particularly prone to this but we see it in any breed of dog, and I see it often in Beagles (including my own Beagle). So, when the back flexes and extends, instead of the jelly compressing and expanding, this chalky substance gets squeezed - and it does not compress, but instead it extrudes out and you thus get a herniated disc.
Treatment for IVDD often involves anti-inflammatories, pain killers and/or steroids. The goal is to decrease the swelling which in turn decreases the pain and improves nerve function. Sometimes, however, they are not enough and the dog becomes paralyzed. In these situations, surgery can be done to go in and remove the disc material that is pressing on the spine. This is called "decompression" surgery.
In order to determine what is appropriate treatment for your dog, a veterinarian needs to perform a very thorough neurological examination. The vet looks for neurological deficits such as a delay in turning the back foot over if it is turned so the top of the foot is on the ground instead of the pads, while the dog is in a standing position.
The vet also looks for "purposeful movement" which is a stepping motion of the hind legs when the vet supports the dog’s weight so the legs can swing freely. There are a number of other neurological tests the vet does to test reflexes. Also, the vet manipulates each vertebrae in a way to find where there may be pain.
Here are some videos of a vet doing a neurological exam:
Here they are checking for PAIN and explain paralysis:
Often, if a painful area is located, the vet will recommend x-rays to look for a compression between the vertebrae. This confirms the diagnosis.
The prognosis for each patient depends on the symptoms, the results of the neurological examination, how long the problem has been present, and how the dog responds to treatment.
I am concerned about your dog because it sounds like he is in pain.
I would strongly recommend a visit to your veterinarian as soon as possible so that he can be assessed and be treated.
Here are some links with more information:
Until you can get him to the vet, please keep him as quiet as possible, with just short visits outside to do potty business. Carry him up and down the stairs, as this is when the back flexes and extends the most, and further damage is most likely.
In terms of whether to go see an emergency vet today or not, you may wish to consider trying a couple of things at home first.
1. You may be able to give some aspirin (as long as he has no history of kidney problems, stomach problems and is not on other medications).
Please use Buffered Aspirin if you have it, and give it with a piece of bread or something low-fat to eat, not on an empty stomach. Enteric coated aspirin is not well absorbed by dogs, so this is not recommended.
Here are links that tell you about it, with precautions and dose:
I would suggest that you give just ONE dose today to keep him comfortable until you see your vet tomorrow, unless you are able to see the vet today. If you give more than one dose, it will prevent your vet from using more effective drugs tomorrow. Also, do mention to your vet that you gave it.
2. Also, many people with back pain report that a warm compress is soothing, and your dog may appreciate that too. You can do this by making a wet towel compress. Place a small wet towel, folded into a zip-lok bag (unzipped!) and heat for about 2 minutes in the microwave. Remove and press all the air out. Make sure it is not too hot! You may want to put another towel around it, and then gently place over your dog's back.
If your Beagle is not finding any relief with these measures, and is unable to settle down, you may need to see an emergency vet today. If, however, he does seem more comfortable, please do get him in to see his family veterinarian promptly tomorrow morning! See a vet immediately if he deteriorates in any way (is taking longer to correct his foot when upside down - check BOTH hind feet every 2 hours).