I have a standard DC power supply adapter, similar to this: Adapter which is center positive; I also have 9V battery clips which are center negative.
Is there an easy way I can use both (one at a time) in the same circuit, is there some kind of cheap converter I can use or (attempt to) make?
If you are looking to convert centre positive to centre negative then these are useful with plug and socket connected back to back and the connections crossed
Thanks, these look handy; I'll get some ordered.
I'm curious though, could I also do it myself:
a) if i checked and knew what the center pin was and just add a switch to swap pos/neg lines.
b) to expand on the idea, is it possible to do this automatically, e.g so any barrel can be plugged in and the circuit automatically switches?
All round DC power jacks have I seen were "center positive". I believe that this is "standard".
I suppose that your battery "center negative" jack is just incorrectly assembled and should be reverted.
Using both "center negative" and "center positive" similar connectors will surely lead to the fact that at some point you mix them up and burn the equipment.
That's why all connectors use the same polarity and there are no adapters for reversing plugs.
Then why is my Brother label printer centre negative ?
Why do some countries drive on the wrong side of the road? 
Is it have the same geometry (like 5.5x2.1) as center positive?
Yes, exactly the same geometry and as I don't have a matching power supply I use a pair of back to back plugs/sockets as I described earlier
Don't you think it's an unsafe engineering design?
Mixed centre positive and negative used to be more prevalent and it is, of course, perfectly safe if you only use the supplied power supply with each device. There is now a centre positive convention but it is a mistake waiting to happen
My converter is kept with the printer and, if nothing else, acts as a reminder that there is something odd about it
It was interesting to know what happens.
However, I still think that in the OP's case, he has some sort of mistake...
You could use a bridge rectifier at the input of your circuit. This would probably drop around 1.5V off the 9V, and if the current is high, e.g. 1A, could get quite hot. But if the current is low, it would be fine. You could use 4x 1N4001 diodes or similar.
You should know that PP3 size 9V batteries are not a great choice for many Arduino circuits. Their capacity is very low and the internal resistance is high, leading to a large voltage drop if more than a small current is drawn from them.
I have an Etón radio that has a center negative barrel power jack. What's weird is it has a "phone charger" barrel jack with the standard center positive. It also takes a 3S NiMH battery pack with a connector of the opposite polarity to all the replacement batteries I can find (the only replacement batteries I found with the same connector are for a VTech cordless phone). It smells very much like vendor lock-in to me (i.e., they want to be the exclusive sellers of cables and adapters).
I've been tempted to swap the polarity on both the power jack and the battery connector and replace the phone charger barrel jack with a more contemporary USB C jack.
Edit to add: rather than vendor lock-in, it could also be that the radio is so old (from 2005, it seems) that phone charger cables and battery connectors weren't nearly as standardized as they are today.
They have them ready-made:
Perfect thanks @dougp, I'll order that as well 
I fear that a "Universal" solution is an accident waiting to happen. The polarity is reversed merely by plugging the 2 pins of the tips so it is easy to make a mistake
@UKHeliBob , is this the same thing?
Not sure if a mistake as such, maybe.
I have a dc adapter (I think it came with an Elegoo kit, some time later I wanted some battery clips to use with these: https://www.amazon.co.uk/dp/B07KYBRJXQ, I probably should have got different types suitable for prototype boards 
Some time even later I thought it might be good to use both the adapter and battery clips on the same board, and then realized about the polarity, which is were the post came from.
Learnt a lot from the replies already whcih is great!
Bridge rectifier - another thing on my list to learn about, thanks.
I wouldn't use this for Arduino (long-term), it's more of a learning excercise.
Most guitar effect pedals use center negative 9V power supply....
Thanks @al1_24 , I didn't even know it was guitar related