Find next record where status field is different from current

You are on the right way. All you need is the LEFT JOIN of the 'Switched ON' part with the 'Switched OFF' part on equal row numbers.

with Events as (
  select 'Switched ON' as ActionTaken, 1 as ID union all -- 3
  select 'Switched ON', 2 union all -- 6
  select 'Switched OFF', 3 union all
  select 'Switched ON', 4 union all -- 7
  select 'Switched ON', 5 union all -- 8
  select 'Switched OFF', 6 union all
  select 'Switched OFF', 7 union all
  select 'Switched OFF', 8 union all
  select 'Switched On', 9 union all -- 10
  select 'Switched OFF', 10 union all
  select 'Switched On', 11 union all -- 12
  select 'Switched OFF', 12
), E as (
  select
    *, row_number() over(partition by ActionTaken order by ID) as rn
  from Events
)
select
  a.ActionTaken, a.ID, b.ID
from E as a
left join E as b
  on a.ActionTaken = 'Switched ON' and
     b.ActionTaken = 'Switched OFF' and
     a.rn = b.rn
order by a.ID, a.ActionTaken;

Output:

+--------------+----+------+
| ActionTaken  | ID |  ID  |
+--------------+----+------+
| Switched ON  |  1 | 3    |
| Switched ON  |  2 | 6    |
| Switched OFF |  3 | NULL |
| Switched ON  |  4 | 7    |
| Switched ON  |  5 | 8    |
| Switched OFF |  6 | NULL |
| Switched OFF |  7 | NULL |
| Switched OFF |  8 | NULL |
| Switched On  |  9 | 10   |
| Switched OFF | 10 | NULL |
| Switched On  | 11 | 12   |
| Switched OFF | 12 | NULL |
+--------------+----+------+

Test it online with SQL Fiddle.