There is one way I know how to achieve a strictly* one-to-one relationship without using triggers, computed columns, additional tables, or other 'exotic' tricks (only foreign keys and unique constraints), with one small caveat.
I will borrow the chicken-and-the-egg concept from the accepted answer to help me explain the caveat.
It is a fact that either a chicken or an egg must come first (in current DBs anyway). Luckily this solution does not get political and does not prescribe which has to come first - it leaves it up to the implementer.
The caveat is that the table which allows a record to 'come first' technically can have a record created without the corresponding record in the other table; however, in this solution, only one such record is allowed. When only one record is created (only chicken or egg), no more records can be added to any of the two tables until either the 'lonely' record is deleted or a matching record is created in the other table.
Solution:
Add foreign keys to each table, referencing the other, add unique constraints to each foreign key, and make one foreign key nullable, the other not nullable and also a primary key. For this to work, the unique constrain on the nullable column must only allow one null (this is the case in SQL Server, not sure about other databases).
CREATE TABLE dbo.Egg (
ID int identity(1,1) not null,
Chicken int null,
CONSTRAINT [PK_Egg] PRIMARY KEY CLUSTERED ([ID] ASC) ON [PRIMARY]
) ON [PRIMARY]
GO
CREATE TABLE dbo.Chicken (
Egg int not null,
CONSTRAINT [PK_Chicken] PRIMARY KEY CLUSTERED ([Egg] ASC) ON [PRIMARY]
) ON [PRIMARY]
GO
ALTER TABLE dbo.Egg WITH NOCHECK ADD CONSTRAINT [FK_Egg_Chicken] FOREIGN KEY([Chicken]) REFERENCES [dbo].[Chicken] ([Egg])
GO
ALTER TABLE dbo.Chicken WITH NOCHECK ADD CONSTRAINT [FK_Chicken_Egg] FOREIGN KEY([Egg]) REFERENCES [dbo].[Egg] ([ID])
GO
ALTER TABLE dbo.Egg WITH NOCHECK ADD CONSTRAINT [UQ_Egg_Chicken] UNIQUE([Chicken])
GO
ALTER TABLE dbo.Chicken WITH NOCHECK ADD CONSTRAINT [UQ_Chicken_Egg] UNIQUE([Egg])
GO
To insert, first an egg must be inserted (with null for Chicken). Now, only a chicken can be inserted and it must reference the 'unclaimed' egg. Finally, the added egg can be updated and it must reference the 'unclaimed' chicken. At no point can two chickens be made to reference the same egg or vice-versa.
To delete, the same logic can be followed: update egg's Chicken to null, delete the newly 'unclaimed' chicken, delete the egg.
This solution also allows swapping easily. Interestingly, swapping might be the strongest argument for using such a solution, because it has a potential practical use. Normally, in most cases, a one-to-one relationship of two tables is better implemented by simply refactoring the two tables into one; however, in a potential scenario, the two tables may represent truly distinct entities, which require a strict one-to-one relationship, but need to frequently swap 'partners' or be re-arranged in general, while still maintaining the one-to-one relationship after re-arrangement. If the more common solution were used, all data columns of one of the entities would have to be updated/overwritten for all pairs being re-arranged, as opposed to this solution, where only one column of foreign keys need to be re-arranged (the nullable foreign key column).
Well, this is the best I could do using standard constraints (don't judge :) Maybe someone will find it useful.