I am trying to understand some of the subtleties of changing the order of suffixes appied to verbs......can someone possibly tell me the difference between the following two words???

başladıysam
başlasaydım

Both are correct according to WinMekMak
Although I am still failing to understand why either require a y buffer consonant :-S

The y buffer is included because historically the narrative tenses was build with "idi" and not "-di" which is the same as past tense.

Narrative tenses are on the pink table on tenses page:
http://www.turkishclass.com/forumTitle_6_2565

So historically 'başlasaydım' was 'başlasa idim', like that as two words. Still in some old sources it is possible to see the "idim" but in today's language it is omited.

A google search will still show a few entries for "bakar idim" or "gelir idim" but of course except original historical works it should be 'gelirdim' and 'bakardım'.

'Başlasaydım' is the Narrative Auxiliaries of Conditional Tenses. The -dı here is not the past tense suffix but the narrative tense suffix. Because the Narrative Auxiliary changed from -idi to -di and this is exactly the same as the past tense suffix there has been some confusion on this issue.

'Başlasaydım' means 'if I had started' (I have not started and now its too late). It is the unreal condition.

Example:
Geçen yıl Fransızca öğrenmeye başlasaydım şimdi konuşuyor olurdum.
If I had started learning French last year, now I would be talking.

As you see in the table on tenses page, almost all tenses have a narrative version except the past tense. If it had existed it todays Turkish it would look like 'geldiydim', 'yaptıydım', gördüydüm'. Some poeple use the narrative of past tense but it is replaced now with narrative of reported past tense 'gelmiştim', 'yapmıştım', görmüştüm'. So our Turkish members should not say 'geldiydim' and should say 'gelmiştim'.

'başladıysam' is the Conditional Auxiliary of Simple Past Tense and means 'if I have started'.

Example:
Sinirlenmeye başladıysam, bunun bir nedeni var.
If I have started to become angry, it is for a reason.

The Conditional Auxiliaries have also changed in the history of our language. In 'başladıysam' you see also the buffer 'y'.

"başladı isem" or "başladım ise" are omited versions. Başlamak is not as common as gelmek or yapmak. So if you check a search engine you will see some entries for "geldi isem", "geldim ise", "yaptı isem", and "yaptım ise". Instead these omited versions başladıysam, geldiysem, yaptıysam should be used.

Never heard about "Imperfect Modifier".

"Narrative Auxiliaries" is related to the word "narrative". You are talking about a certain time in the past. In general it is a good idea to imagine "Narrative Auxiliary" as a sentence which would start as "once" or "at the time".

I will make a few translations from tenses thread, from the pink table.
http://www.turkishclass.com/forumTitle_6_2565

Narrative Auxiliary of Simple present Tense
(mostly unreal, rarely real)
gelirdim > I would have come
Arasaydın gelirdim.
If you had called me I would have come.

Narrative Auxiliary of Present continuous Tense
(real)
geliyordum > I was coming

Narrative Auxiliary of Reported past Tense
(unreal or real)
gelmiştim > I had come, I would have come

Narrative Auxiliary of Future Tense
(unreal)
gelecektim > I was going to come

Narrative Auxiliary of Conditional Tense
(unreal)
gelseydim > If I had come

Narrative Auxiliary of Necessitive Tense
(unreal)
gelmeliydim > I should have come

I hope this helps. These translations are a bit tricky. I haven't read them somewhere but I feel like I had made smillar translations before. It's a shame some good information is here and there in different threads. Bye.

Incidentally, WinMekMak gives exactly the same meaning (If I began) to both:

başladıysam
başlasaydım

You are right. I just checked winmekmak and it is indeed "imperfect modifier" in this program.

I also compared my translation. Well, there are tiny differences. Not so bad.

I love WinMekMak, but I never trusted the translations too much. Sometimes there even are none..