Well, since we want the days to be on the same weekday:
We want a precise number of weeks to have elapsed, i.e. 0 (mod 7) days. We have the following lengths for the months (total and mod 7, respectively):
- 31, 3
- 28 or 29, 0 or 1 (depending on whether it's a leap year)
- 31, 3
- 30, 2
- 31, 3
- 30, 2
- 31, 3
- 31, 3
- 30, 2
- 31, 3
- 30, 2
- 31, 3 (although this isn't really important, since we won't progress past this.)
Then:
It's easy to determine whether January xth is on the same day as October xth, by simply adding those numbers (mod 7). However, reversing the date complicates things a bit. We can get around this by using the difference in day and month, and adding/subtracting that from the found number.
That is, January 3rd becomes March 1st. January -> March gives a 3 or 4 day difference. March 1st is 2 days before March 3rd, so we subtract 2, giving 1 or 2. Thus, as 3/1/18 (dd/mm/yy) is on Wednesday, and 2018 is not a leap year, 1/3/18 (dd/mm/yy) is a Thursday. (Yes, I did check my calendar here to see if my approach actually works.)
Finishing up:
Since the difference between the same day in the other month, and the actual reversed date is constant for each month, we can combine those:
Jan -> Feb: 3-1 = 2 days difference.
Jan -> Mar: 3 or 4 -2 = 1 or 2 days difference (as before).
Jan -> Apr: 6 or 7 -3 = 3 or 4 days difference.
Jan -> May: 8 or 9 - 4 = 4 or 5 days difference.
Jan -> Jun: 11 or 12 -5 = 6 or 7 days difference - We have an ambiguous date in leap years!
Jan -> Jul: 13 or 14 -7 = 7 or 8 days difference - Ambiguous in non-leap years.
We can simply continue this approach. If we only consider dates where the reverse is later in the year (i.e. 3/1 will be considered in January, so 1/3 not in March), we get:
Jan
: 1 in leap years - June, 2 in non-leap years - July and November.
Feb
: 2 in leap years (Mar/Dec), 1 in non-leap years (Aug)
Mar
: 1 (Dec) (Note: Leap years don't matter any more, since we're past February)
Apr
: 0
May
: 1 (Sep)
Jun
: 0
Jul
: 1 (Nov)
All the other months don't have ambiguous dates with months later than that one.
EDIT: Here's a picture of an excel sheet with my approach (made after doing all of this by hand...):

The green cells indicate matching months. Out of view I have a list with the month lengths mod 7, where I can easily change Feb from 0 to 1 for leap years.
Giving a total of:
6 ambiguous dates in both leap and non-leap years.
I surely hope I didn't make any mistakes counting (Edit: January was actually wrong. Correcter that, which nicely ensures we have the same number in every year :D). I checked one or two of them which were correct, but I won't give any guarantees. Also, I noticed I got rather annoyed by the fact that July and August both have 31 days, rather than simply alternating like the /normal/ months.
WEEKDAY(DATE(2016,A1,B1))=WEEKDAY(DATE(2016,B1,A1))
$\endgroup$