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I don't know of anything built in to do this, but here's one possible method. I think it requires slightly different definitions of R1 and R2, but maybe you can work something out if you need to keep your definitions:

sage: S1 = ZZ['x,y']
sage: S2 = ZZ['y,x']
sage: a = S1.gen(0) * (S1.gen(1) + 1)
sage: a
x*y + x
sage: a.dict()
{(1, 0): 1, (1, 1): 1}
sage: S2(a.dict())
y*x + y

So I guess with your original R1 and R2:

sage: R1 = ZZ['y']['x']
sage: R2 = ZZ['x']['y']
sage: a = R1.base_ring().gen(0) * (R1.gen(0) + 1)
sage: a
y*x + y
sage: a.dict()
{0: y, 1: y}
sage: R2({p: a.dict()[p].subs(R2.base_ring().gen(0)) for p in a.dict()})
x*y + x

I don't know of anything built in to do this, but here's one possible method. I think it requires slightly different definitions of R1 and R2, but maybe you can work something out if you need to keep your definitions:

sage: S1 = ZZ['x,y']
sage: S2 = ZZ['y,x']
sage: a = S1.gen(0) * (S1.gen(1) + 1)
sage: a
x*y + x
sage: a.dict()
{(1, 0): 1, (1, 1): 1}
sage: S2(a.dict())
y*x + y

Edit: sorry, this doesn't do what you want. You should modify the dictionary appropriately:

sage: d = {(p[1], p[0]):a.dict()[p] for p in a.dict()}
sage: S2(d)
y*x + x

So I guess with your original R1 and R2:

sage: R1 = ZZ['y']['x']
sage: R2 = ZZ['x']['y']
sage: a = R1.base_ring().gen(0) * (R1.gen(0) + 1)
sage: a
y*x + y
sage: a.dict()
{0: y, 1: y}
sage: R2({p: a.dict()[p].subs(R2.base_ring().gen(0)) for p in a.dict()})
x*y + x

I don't know of anything built in to do this, but here's one possible method. I think it requires slightly different definitions of R1 and R2, but maybe you can work something out if you need to keep your definitions:

sage: S1 = ZZ['x,y']
sage: S2 = ZZ['y,x']
sage: a = S1.gen(0) * (S1.gen(1) + 1)
sage: a
x*y + x
sage: a.dict()
{(1, 0): 1, (1, 1): 1}
sage: S2(a.dict())
y*x + y

Edit: sorry, this doesn't do what you want. You should modify the dictionary appropriately:

sage: d = {(p[1], p[0]):a.dict()[p] for p in a.dict()}
sage: S2(d)
y*x + x

So I guess with your original R1 and R2:

sage: R1 = ZZ['y']['x']
sage: R2 = ZZ['x']['y']
sage: a = R1.base_ring().gen(0) * (R1.gen(0) + 1)
sage: a
y*x + y
sage: a.dict()
{0: y, 1: y}
sage: R2({p: a.dict()[p].subs(R2.base_ring().gen(0)) for p in a.dict()})
x*y + x

Edit: There should be a similar way to modify the dictionary here, too...