Actually most cars do NOT have syncros in reverse. It is very common. The best practice is to always go into 1st and then go into reverse. This is a technique I learned many years ago and it still serves well today. When you are idling in neutral and put your foot on the clutch and then go quickly right into reverse the input shaft of the transmission is still spinning down and with no syncro you'll grind momentarily. You can either wait a second or two after pushing in the clutch before going to reverse or put the clutch in, go quickly into 1st before going into reverse. This will stop the input shaft from spinning using the 1st gear syncro. Then you can go nice and easy into reverse. It also works on cars that are sometimes quirky going all the way into reverse. The gears don't always line up exactly and it won't go completely into reverse. The trick there is to go back to neutral, let the clutch out to spin the input shaft of the trans, push the clutch back in, go into first, and then to reverse again. Once you are in the habit of "1st-Reverse" it's really no trouble at all and I have taught all my daughters the technique. Yes, I force them all to learn and drive sticks. At first they don't like me but then they're pretty happy to know since nowadays you can get manual transmission cars for far less than automatics because nobody knows how to drive them!
If a transmission is having issues going into ALL gears then you'll want to take a look at clutch adjustment to make sure the clutch is releasing completely. On hydraulic clutch acutators you can get air in the system which will prevent the clutch from disengaging completely, causing grinding in all the gears because the input shaft of the trans is still being driven by the engine somewhat. On cable clutch actuators it's the same but you will want to check the cable adjustment.